Long-lived particle phenomenology in the 2HDM+$a$ model

Higgs decays displaced from the primary interaction vertex represent a striking experimental signature that is actively searched for by the ATLAS, CMS and LHCb collaborations. We point out that signals of this type appear in the context of the 2HDM+$a$ model if the mixing angle $\theta$ of the two CP-odd weak spin-0 eigenstates is tiny and the dark matter (DM) sector is either decoupled or kinematically inaccessible. Utilising two suitable benchmark scenarios, we determine the constraints on the parameter space of the 2HDM+$a$ model that are set by the existing LHC searches for long-lived particles (LLPs) in Higgs decays. We find that depending on the precise mass spectrum of the spin-0 states, mixing angles $\theta$ in the ballpark of a few $10^{-8}$ to $10^{-5}$ can be excluded based on LHC Run II data. This finding emphasises the unique role that searches for displaced signatures can play in constraining the parameter space of the 2HDM+$a$ model. The ensuing DM phenomenology is also discussed. In particular, we show that parameter choices leading to an interesting LLP phenomenology can simultaneously explain the DM abundance observed in today's Universe.