  The attractive inverse square potential arises in a number of physical
problems such as a dipole interacting with a charged wire, the Efimov effect,
the Calgero-Sutherland model, near-horizon black hole physics and the optics of
Maxwell fisheye lenses. Proper formulation of the inverse-square problem
requires specification of a boundary condition (regulator) at the origin
representing short-range physics not included in the inverse square potential
and this generically breaks the Hamiltonian's continuous scale invariance in an
elementary example of a quantum anomaly. The system's spectrum qualitatively
changes at a critical value of the inverse-square coupling, and we here point
out that the transition at this critical potential strength can be regarded as
an example of a $\mathcal{PT}$ symmetry breaking transition. In particular, we
use point particle effective field theory (PPEFT), as developed by Burgess et
al [J. High Energy Phys., 2017(4):106, 2017], to characterize the
renormalization group (RG) evolution of the boundary coupling under rescalings.
While many studies choose boundary conditions to ensure the system is unitary,
these RG methods allow us to systematically handle the richer case of
nonunitary physics describing a source or sink at the origin (such as is
appropriate for the charged wire or black hole applications). From this point
of view the RG flow changes character at the critical inverse-square coupling,
transitioning from a sub-critical regime with evolution between two real,
unitary fixed points ($\mathcal{PT}$ symmetric phase) to a super-critical
regime with imaginary, dissipative fixed points ($\mathcal{PT}$ symmetry broken
phase) that represent perfect-sink and perfect-source boundary conditions,
around which the flow executes limit-cycle evolution.
