  The Efimov phenomenon manifests itself as an emergent discrete scaling
symmetry in the quantum three-body problem. In the unitarity limit, it leads to
an infinite tower of three-body bound states with energies forming a geometric
sequence. In this work, we study the evolution of these so-called Efimov states
using relativistic scattering theory. We identify them as poles of the
three-particle $S$ matrix and trace their trajectories in the complex energy
plane as they evolve from virtual states through bound states to resonances. We
dial the scattering parameters toward the unitarity limit and observe the
emergence of the universal scaling of energies and couplings -- a behavior
known from the non-relativistic case. Interestingly, we find that Efimov
resonances follow unusual, cyclic trajectories accumulating at the three-body
threshold and then disappear at some values of the two-body scattering length.
We propose a partial resolution to this "missing states" problem.
