Decay of Z into Three Pseudoscalar Bosons

We consider the decay of the $Z$ boson into three pseudoscalar bosons in a general two-Higgs-doublet model. Assuming $m_A$ to be very small, and that of the two physical neutral scalar bosons $h_1$ and $h_2$, $A$ only couples to $Z$ through $h_1$, we find the $Z\to AAA$ branching fraction to be negligible for moderate values of $\tan \beta \equiv v_2\mathrm{/}{v}_1$, if there is no $\lambda_5 (\Phi_1^\dagger \Phi_2)^2 + h.c.$ term in the Higgs potential; otherwise there is no absolute bound but very large quartic couplings (beyond the validity of perturbation theory) are needed for it to be observable.