The cost of an ALP solution to the neutral $B$-anomalies

The neutral anomalies in $B$ decays are analysed in terms of the tree-level exchange of an axion-like-particle (ALP), within the effective field theory framework. The complete two-dimensional parameter space for ALP couplings to electrons and muons is explored. The solutions to $R_K$ and to the two energy bins of $R_{K^\ast}$ are confronted with the impact of ALP exchange on other observables (meson oscillations, leptonic and semileptonic decays of $B$ mesons including searches for new resonances, astrophysical constraints), as well as with the theoretical domain of validity of the effective theory. Solutions based on ALPs heavier than $B$ mesons, or lighter than twice the muon mass, are shown to be excluded. In contrast, the exchange of on-shell ALPs provides solutions to $R_K$ and/or $R_{K^\ast}$ within $2\sigma$ sensitivity which are technically compatible with those constraints. Furthermore, a ''golden ALP mass'' is identified at the frontier between the two energy bin windows of $R_{K^\ast}$, which could simultaneously explain these two $R_{K^\ast}$ anomalies together with $R_K$; this calls for the convenience of different energy binning which would easily clear up this (unlikely) possibility. The impact of smearing on data analysis is also discussed. When loop effects are taken into account, the solutions found can be in addition compatible with the data on the $g-2$ of the electron but not simultaneously with those on the $g-2$ of the muon. Furthermore, loop effects may require fine-tunings of some coupling values.