  Black holes have sometimes been described as astrophysical particle
accelerators because finite energy particles can collide near the horizon with
divergent center-of-mass (CM) energy. The collisions are classified by the
radial motion of the constituent particles at the moment of collision, with
each class exhibiting a distinct near-horizon behavior. Divergence in the CM
energy is sourced by the difference in the rate at which the collision radius
approaches the horizon and the rate at which a particle's angular momentum is
tuned to a critical value set by the superradiant bound. To produce a high
energy collision around a near-extremal BH, at least one particle must approach
criticality slower than the collision radius approaches the horizon. When both
particles are ingoing or outgoing, it is additionally required that the
particles approach criticality at different rates. Using a novel multi-scaling
limit, we calculate the explicit form for the divergent CM energy.
