An "alien" called Oosterhoff dichotomy?

In this letter we investigate the origin of the Oosterhoff dichotomy, considering recent discoveries related to several ancient merging events of external galaxies with the Milky Way (MW). In particular, we aim to clarify if the subdivision in Oosterhoff type of Galactic Globular Clusters (GGCs) and field RR Lyrae (RRLs) could be traced back to one or more ancient galaxies that merged with the MW in its past. To this purpose, we first explored the association of GGCs with the past merging events according to different literature studies. Subsequently we compiled positions, proper motions and radial velocity for 10,138 field RRLs variables from the $Gaia$ Data Release 3. To infer the distances, we adopted the $M_G$--[Fe/H] relation, with [Fe/H] values estimated through empirical relationships involving the individual periods and Fourier parameters. We then calculated the orbits and the integrals of motions (IoM) using the Python library Galpy for the whole sample. By comparing the location of the field RRLs in the energy-angular momentum diagram with that of the GGCs we assign their likely origin. Finally, we discriminate from the $Gaia$ G-band light curves the Oosterhoff type of our sample of RRL stars based on their location in the Bailey diagram. The analysis of the Bailey diagrams for Galactic RRLs stars and GGCs associated with \textit{In-Situ} vs \textit{Accreted} halo origin shows remarkable differences. The \textit{In-Situ} sample displays a wide range of metallicities with a continuous distribution and no sign of Oosterhoff dichotomy. Conversely, the \textit{Accreted} RRLs clearly shows the Oosterhoff dichotomy and a significantly smaller dispersion in metallicity. Our results suggest that the Oosterhoff dichotomy was imported into the MW by the merging events that shaped the Galaxy.