Probing Muonic Forces and Dark Matter at Kaon Factories

Rare kaon decays are excellent probes of light, new weakly-coupled particles. If such particles $X$ couple preferentially to muons, they can be produced in $K\to \mu \nu X$ decays. In this letter we evaluate the future sensitivity for this process at NA62 assuming $X$ decays either invisibly or to di-muons. Our main physics target is the parameter space that resolves the $(g-2)_\mu$ anomaly, where $X$ is a gauged $L_\mu-L_\tau$ vector or a muon-philic scalar. The same parameter space can also accommodate dark matter freeze out or reduce the tension between cosmological and local measurements of $H_0$ if the new force decays to dark matter or neutrinos, respectively. We show that for invisible $X$ decays, a dedicated single muon trigger analysis at NA62 could probe much of the remaining $(g-2)_\mu$ favored parameter space. Alternatively, if $X$ decays to muons, NA62 can perform a di-muon resonance search in $K\to 3 \mu \nu$ events and greatly improve existing coverage for this process. Independently of its sensitivity to new particles, we find that NA62 is also sensitive to the Standard Model predicted rate for $K \to 3\mu \nu$, which has never been measured.