Wannier threshold law and the classical-quantum correspondence in three-body Coulomb breakup

We derive the classical threshold law for the breakup of three charged particles interacting via long-range Coulomb forces for the case of a charge ratio of $1/4$ between the particles involved. The Wannier exponent in canonical Wannier theory is known to diverge in this case. We find that the classical threshold law, which is proportional to exp(-λ/√E), shows exponential suppression of the ionization probability at threshold. We thus identify the possibility that exponential behavior in breakup processes, typically attributed to quantum-mechanical tunneling, can arise as a completely classical dynamical effect. We show that in the limit of zero energy, the behavior of the cross section is characterized by a classical to quantum transition. We discuss the regime of parameters in which this transition occurs and the possibility of an experimental observation of this transition.