Classical dynamics of three-body systems in the Efimov potential

Abstract

We investigate the classical nonrelativistic dynamics of three bodies in Efimov's $\ensuremath{-}1/{R}^{2}$ potential in three spatial dimensions, with a view towards semiclassical quantization and insight into the geometry of Efimov states. Without the short-distance cutoff, these dynamics are superintegrable, which allows an exact integration of the equations of motion for arbitrary initial conditions. We show that periodic orbits necessarily lead to exactly vanishing binding energy of the bound states, in disagreement with Efimov's quantum-mechanical results. A scaling anomaly demands that the quantum dynamics of three bodies in this potential be augmented by additional boundary conditions affecting all three particles at the short-distance cutoff point, i.e., near the triple collision. We discuss the inherent difficulties in the definition of appropriate three-body boundary conditions in three spatial dimensions and briefly discuss their consequences for (quasi)periodic orbits. Consequently, the classical orbits corresponding to Efimov states cannot be exactly periodic, but must have a finite timescale (lifetime), associated with the time it takes the system's hyperradius to fall to zero, or to the cutoff value, which is typically much longer than the (quasi)period of the hyperangular motion. The scaling properties of the lifetime are in agreement with the quantum-mechanical predictions of the half-life (width) of Efimov states surrounded by an ultracold gas. Detailed spatiotemporal evolution of the system is generally unpredictable beyond the three-body collision point, even though global conservation laws ensure that the system's hyperradius must be periodic.