K006 $K_L \to \mu^+ \mu^-$

In warped or anarchic-flavor models this channel is relevant through the same flavor-changing \(s\to d\) neutral-current structures that appear in rare kaon phenomenology: modified \(Z\) couplings, heavy gauge-boson penguins, and box topologies. It is less clean than \(K\to\pi\nu\bar\nu\), because the measured branching fraction is almost saturated by long-distance two-photon physics, but it is still a useful diagnostic for large semileptonic short-distance effects. The Isidori-Unterdorfer dispersive analysis quoted in the sidecar gives a conservative bound \(\mathcal{B}(K_L\to\mu^+\mu^-)_{\rm short}<2.5\times10^{-9}\), which is the right order-of-magnitude target for any future catalog-to-code translation.

No newer high-precision \(K_L\to\mu^+\mu^-\) branching-fraction measurement was found beyond the PDG average used above. The main post-2008 movement is theoretical: D'Ambrosio-Kitahara identified direct-CP and \(K_L\)-\(K_S\) interference handles for \(K\to\mu^+\mu^-\); Dery-Ghosh-Grossman-Schacht showed that time-dependent information can isolate a clean short-distance \(K_S\) observable with hadronic uncertainty below \(1\%\); and Chao-Christ developed a lattice-QCD framework for the long-distance two-photon amplitude, with a quoted \(10\%\) target accuracy. These papers do not replace the PDG branching fraction, but they sharpen how this family of modes could become a cleaner short-distance probe after the CFW-era treatment.

As a direct rate constraint this is long-distance-limited. New physics with large scalar or pseudoscalar lepton operators would require separate treatment; the short-distance interpretation above is aimed at vector/axial semileptonic operators generated by \(Z\)-penguins and electroweak boxes. A production constraint should therefore use either a conservative dispersive allowance or a future lattice-assisted SM subtraction, not the total PDG branching fraction as a clean short-distance observable.

NO. A targeted grep across quarkConstraints/, qcd/, flavorConstraints/, neutrinos/, yukawa/, warpConfig/, solvers/, scanParams/, and tests/, excluding notebooks and binary-like assets, found no \(K_L\to\mu^+\mu^-\), dimuon, or \(s\to d\ell^+\ell^-\) implementation. The nearest kaon hit is quarkConstraints/deltaf2.py:615, which defines the \(K_L-K_S\) mass difference for \(\Delta F=2\) mixing, not this decay.

HIGH. Implementing K006 would need a new rare-kaon semileptonic observable, short-distance \(s\to d\mu^+\mu^-\) matching, and an explicit treatment of the long-distance two-photon dispersive amplitude or a conservative external bound. Those ingredients are not present in the audited \(\Delta F=2\) code path.