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

A giant planet has been recently resolved at a projected distance of 730 au from the tight pair of young ($\sim$ 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. We aim at studying 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). 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. 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. 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.