Origin of the wide-orbit circumbinary giant planet HD 106906 A dynamical scenario and its impact on the disk

Context. A giant planet has been recently resolved at a projected distance of 730 au from the tight pair of young (similar to 13 Myr) intermediate-mass stars HD 106906AB in the Lower Centaurus Crux (LCC) group. The stars are surrounded by a debris disk which displays a ring-like morphology and strong asymmetries at multiple scales. Aims. We aim to study the likelihood of a scenario where the planet formed closer to the stars in the disk, underwent inward disk-induced migration, and got scattered away by the binary star before being stabilized by a close encounter (fly-by). Methods. We performed semi-analytical calculations and numerical simulations (Swift_HJS package) to model the interactions between the planet and the two stars. We accounted for the migration as a simple force. We studied the LCC kinematics to set constraints on the local density of stars, and therefore on the fly-by likelihood. We performed N-body simulations to determine the effects of the planet trajectories (ejection and secular effects) onto the disk morphology. Results. The combination of the migration and mean-motion resonances with the binary star (often 1:6) can eject the planet. Nonetheless, we estimate that the fly-by hypothesis decreases the scenario probability to less than 10(-7) for a derived local density of stars of 0.11 stars/pc(3). We show that the concomitant effect of the planet and stars trajectories induce spiral-features in the disk which may correspond to the observed asymmetries. Moreover, the present disk shape suggests that the planet is on an eccentric orbit. Conclusions. The scenario we explored is a natural hypothesis if the planet formed within a disk. Conversely, its low probability of occurrence and the fact that HD 106906 b shares some characteristics with other systems in Sco-Cen (e.g., HIP 78530, in terms of mass ratio and separation) may indicate an alternative formation pathway for those objects.