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To explain the anomalously large decay rate of $\Sigma^+\to p+\mu^+\mu^-$, itwas proposed that a new mechanism where a light CP-odd pseudoscalar boson of$m_{A_1^0}=214.3$ MeV makes a crucial contribution. Later, some authors havestudied the transition $\pi^0\to e^+e^-$ and $\Upsilon\to \gamma A_1^0$ interms of the same mechanism and their result indicates that with the suggestedmass one cannot fit the data. This discrepancy might be caused by experimentalerror of $\Sigma^+\to p+\mu^+\mu^-$ because there were only a few events.Whether the mechanism is a reasonable one motivates us to investigate thetransitions $\pi^0\to e^+e^-; \eta (\eta^\prime)\to \mu^+\mu^-; \eta_c\to\mu^+\mu^-; \eta_b\to\tau^+\tau^-$ within the same framework. It is noted thatfor $\pi^0\to e^+e^-$, the standard model (SM) prediction is smaller than thedata, whereas the experimental central value of $\eta \to \mu^+\mu^-$ is alsoabove the SM prediction. It means that there should be extra contributions fromother mechanisms and the contribution of $A_1^0$ may be a possible one.Theoretically calculating the branching ratios of the concerned modes, we wouldcheck if we can obtain a universal mass for $A_1^0$ which reconcile thetheoretical predictions and data for all the modes. Unfortunately, we find thatit is impossible to have such a mass with the same coupling $|g_\ell|$.Therefore we conclude that the phenomenology does not favor such a light$A_1^0$, even though a small window is still open.

Study on the effects of the light CP-odd Higgs via the leptonic decays of pseudoscalar mesons