Kaon decays shedding light on massless dark photons

We explore kaon decays with missing energy carried away by a massless dark photon, $\overline\gamma$, assumed to have flavor-changing dipole-type couplings to the $d$ and $s$ quarks. We consider in particular the neutral-kaon modes $K_L\to\gamma\overline\gamma$ and $K_L\to\pi^0\gamma\overline\gamma$ and their $K_S$ counterparts, as well as the charged-kaon channel $K^+\to\pi^+\gamma\overline\gamma$, each of which also has an ordinary photon, $\gamma$, in the final state. In addition, we look at $K_{L,S}\to\pi^+\pi^-\overline\gamma$ and $K^+\to\pi^+\pi^0\overline\gamma$. Interestingly, the same $ds\overline\gamma$ interactions give rise to the flavor-changing two-body decays of hyperons with missing energy and are subject to model-independent constraints that can be inferred from the existing hyperon data. Taking this into account, we obtain branching fractions ${\cal B}(K_L\to\gamma\overline\gamma)$ and ${\cal B}(K_L\to\pi^0\gamma\overline\gamma)$ which can be as high as $10^{-3}$ and $10^{-6}$, respectively, one or both of which may be within the sensitivity reach of the KOTO experiment. Furthermore, we find that ${\cal B}(K^+\to\pi^+\gamma\overline\gamma)$ and ${\cal B}(K^+\to\pi^+\pi^0\overline\gamma)$ are allowed to be maximally of order $10^{-6}$ as well, which may be probed by NA62. Complementarily, the hyperon modes can have rates which are potentially accessible by BESIII. Thus, these ongoing experiments could soon be able to offer significant tests on the existence of the massless dark photon.