Probing QCD Axions or Axion-like Particles in three-body $K$ Decays

Two-body decays like $K \to \pi a$ rank among the most constraining collider probes for new, low-mass, feebly interacting pseudoscalar particles $a$. We explore an alternative class of kaon decay modes, specifically three-body decays to $\pi \pi a$ or $\mu \mu a$. The former occur at tree level, while the latter is loop-suppressed yet accidentally finite. These modes specifically leverage the accurate tracking capabilities at LHCb. We present an estimation approach for the sensitivity achievable in future searches within these channels. Our argument uses the current uncertainty in leading irreducible backgrounds identified for each channel. Our findings suggest that dedicated searches could probe $f_a$ scales between $10^4$ and $10^6$ TeV, highlighting their strong potential. A direct comparison with actual searches, only available in the $K^+ \to \pi^+ \pi^0 a$ channel, supports this conclusion. Finally, we show that, in these searches, reconstruction efficiency maps are such that large efficiencies are naturally aligned with regions of higher yields in Dalitz plots.