  We study the possibility to find the axion-like particles (ALPs) through the
leptonic decays of heavy mesons. There are some deviations between the Standard
Model (SM) predictions of the branching ratios of the leptonic decays of mesons
and the experimental data. This provides some space for the existence of decay
channels where the ALP is one of the products. Three scenarios are considered:
first, the ALP is only coupled to one single charged fermion, namely, the
quark, the antiquark, or the charged lepton; second, the ALP is only coupled to
quark and antiquark with the same strength; third, the ALP is coupled to all
the charged fermions with the same strength. The constraints of the coupling
strength in different scenarios are obtained by comparing the experimental data
of the branching ratios of leptonic decays of $B^-$, $D^+$, and $D_s^+$ mesons
with the theoretical predictions which are achieved by using the Bethe-Salpeter
(BS) method. These constraints are further applied to predict the upper limits
of the leptonic decay processes of the $B_c^-$ meson in which the ALP
participates.
