Axion-like particles as mediators for dark matter: beyond freeze-out

We consider an axion-like particle (ALP) coupled to Standard Model (SM) fermions as a mediator between the SM and a fermionic dark matter (DM) particle. We explore the case where the ALP-SM and/or the ALP-DM couplings are too small to allow for DM generation via standard freeze-out. DM is therefore thermally decoupled from the visible sector and must be generated through either freeze-in or decoupled freeze-out (DFO). In the DFO regime, we present an improved approach to obtain the relic density by solving a set of three stiff coupled Boltzmann equations, one of which describes the energy transfer from the SM to the dark sector. Having determined the region of parameter space where the correct relic density is obtained, we revisit experimental constraints from electron beam dump experiments, rare $B$ and $K$ decays, exotic Higgs decays at the LHC, astrophysics, dark matter searches and cosmology. In particular, for our specific ALP scenario we (re)calculate and improve beam dump, flavour and supernova constraints. Throughout our calculation we implement state-of-the-art chiral perturbation theory results for the ALP partial decay width to hadrons. We find that while the DFO region, which predicts extremely small ALP-fermion couplings, can probably only be constrained by cosmological observables, the freeze-in region covers a wide area of parameter space that may be accessible to other more direct probes. Some of this parameter space is already excluded, but a significant part should be accessible to future collider experiments.