and we present the matrix elements below $$\begin{aligned}
&& \!\!\! \! \!\! \! \! V_{K^+ K^- \to \pi^0 \eta} \!  = \frac{-
\sqrt{3}}{12f^2} \! ( 3s \!\! - \! \! \frac{8}{3}m^2_K \!\! - \!\! \frac{1}{3}m^2_{\pi} \!\! - \!\! m^2_{\eta} ), \\
&& \!\!\! \! \!\! \! \! V_{K^0\bar{K}^0 \to \pi^0 \eta} = - V_{K^+ K^- \to \pi^0 \eta} ,\\
&& \!\!\! \! \!\! \! \! V_{\pi^0 \eta \to \pi^0 \eta} = -\frac{1}{3f^2}m^2_{\pi}, \\
&& \!\!\! \! \!\! \! \! V_{K^+K^- \to K^+K^-} = -\frac{1}{2f^2} s , \\
&& \!\!\! \! \!\! \! \! V_{K^+K^- \to K^0 \bar{K}^0} = -\frac{1}{4f^2} s , \\
&& \!\!\! \! \!\! \! \! V_{K^0 \bar{K}^0 \to K^0 \bar{K}^0} =
-\frac{1}{2f^2} s ,\end{aligned}$$ with $f$ the pion decay constant, $f = 93$ MeV, and $s$ is invariant mass square of the meson-meson system.

The loop function $G$ [@npa] is regularized by means of a cut off. When the $\eta \eta$ channel is explicitly taken into account the cut off needed is smaller than in Ref. [@npa] and we follow [@weihong] where it was taken equal to $q_{\rm max} =
600$ MeV.

Finally, the mass distribution for the decay is given by [^1] $$\label{eq:dGamma}
\frac{d \Gamma}{d M_{\rm 