Black hole with a scalar field as a particle accelerator

We consider stationary axially symmetric black holes with the background scalar field and test particles that can interact with this field directly. Then, particle collision near a black hole can lead to the unbounded energy $$ in the centre of mass frame (contrary to some recent claims in literature). This happens always if one of particles is neutral whereas another one has nonzero scalar charge. Kinematically, two cases occur here. (i) A neutral particle approaches the horizon with the speed of light while the velocity of the charged one remains separated from it (this is direct analogue of the situation with collision of geodesic particles.). (ii) Both particles approach the horizon with the speed almost equal to that of light but with different rates. As a result, in both cases the relative velocity also approaches the speed of light, so that $E_{c.m.}$ $\$becomes unbounded. We consider also a case when the metric coefficient $g_{\phi \phi }\rightarrow 0$ near a black hole. Then, overlap between the geometric factor and the presence of the scalar field opens additional scenarios in which unbounded energy $E_{c.m.}$ is possible as well. We give a full list of possible scenarios of high-energy collisions for the situations considered.