Kaon decay constraints on vector bosons coupled to non-conserved currents

We study rare three- and four-body kaon decays as a probe of light vector and axial-vector bosons coupled to non-conserved currents. We find that searches for $K_L \to \pi^0 \pi^0 (X\to e^+e^-)$ decays constrain the couplings of light $X$ bosons to light quarks to be as small as $\mathcal{O}(10^{-5})$. The charged-pion modes $K^+ \to \pi^+ \pi^0 (X \to e^+e^-)$ and $K_L \to \pi^+ \pi^- (X \to e^+e^-)$ provide weaker limits, but constrain complementary combinations of couplings to the $u$, $d$, and $s$ quarks at the level of $\mathcal{O}(10^{-4})$. Finally, we also find that double emission of $X$ in $K \to \pi XX$ decays can provide yet additional constraints on the parameter space of light $X$ bosons due to a double $(m_K/m_X)^2$ enhancement to the rate. For a 17 MeV boson, these limits add to the known tension between spin-1 bosons coupled to vector and axial-vector currents interpretations of the results of the ATOMKI experiment with meson decay data. Finally, we also comment on negative pion capture on hydrogen and deuterium as a source of light particles and discuss the prospects for testing the 17 MeV boson hypothesis.