$B\rightarrow K + \text{axion-like particles}$: effective versus UV-complete models and enhanced two-loop contributions

An axion-like particle $a$ (ALP) can explain the excess of $B\rightarrow K + \text{invisible}$ events at Belle-II. However, many analyses of ALP scenarios are over-simplified. We revisit the $B\rightarrow K a$ transition rate in a popular minimal and UV complete model with two Higgs doublets (2HDM) and a complex singlet (DFSZ model). To this end we compare our results with previous studies which derived the $\overline{b}sa$ vertex from the $\overline{b}sA$ vertex, where $A$ is the heavy pseudo-scalar of the 2HDM, in terms of an $a-A$ mixing angle. We find this approach to work only at the leading one-loop order, while it fails at the two-loop level. Furthermore, while an approximate $Z_2$ symmetry suppresses the leading-order amplitude by a factor of $1/\tan\beta$, which is the ratio of the two vacuum expectation values of the Higgs doublets, we find the two-loop contribution unsuppressed and phenomenologically relevant for $\tan\beta \gtrsim 5$. We determine the allowed parameter space and underline the importance of better searches for $\Upsilon\rightarrow \gamma+$invisible and for a possible excess in $B\rightarrow K\mu^+\mu^-$. We further study the low-energy axion effective theory which leads to a divergent and basis-dependent amplitude. As a conceptual result, we clarify the ambiguities and identify which low-energy framework is consistent with the DFSZ model.