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The attractive inverse square potential arises in a number of physicalproblems such as a dipole interacting with a charged wire, the Efimov effect,the Calgero-Sutherland model, near-horizon black hole physics and the optics ofMaxwell fisheye lenses. Proper formulation of the inverse-square problemrequires specification of a boundary condition (regulator) at the originrepresenting short-range physics not included in the inverse square potentialand this generically breaks the Hamiltonian's continuous scale invariance in anelementary example of a quantum anomaly. The system's spectrum qualitativelychanges at a critical value of the inverse-square coupling, and we here pointout that the transition at this critical potential strength can be regarded asan example of a $\mathcal{PT}$ symmetry breaking transition. In particular, weuse point particle effective field theory (PPEFT), as developed by Burgess etal [J. High Energy Phys., 2017(4):106, 2017], to characterize therenormalization 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 ofnonunitary physics describing a source or sink at the origin (such as isappropriate for the charged wire or black hole applications). From this pointof 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-criticalregime with imaginary, dissipative fixed points ($\mathcal{PT}$ symmetry brokenphase) that represent perfect-sink and perfect-source boundary conditions,around which the flow executes limit-cycle evolution.

Fall-to-the-centre as a $\mathcal{PT}$ symmetry breaking transition