Singular Potentials and Anomalous Symmetry Breaking

Abstract

This work investigates attractive potentials of the form 1/rN for N ≥ 2 in Quantum Mechanics. For the inverse square potential (N = 2) with sufficiently strong coupling, we examine how classical scale symmetry is broken through the renormalization process, representing the simplest example of a quantum anomaly and touching upon the fundamental question of what symmetries are present in reality. This anomaly is exemplified by the existence of a critical dipole moment required to bind an electron to a polar molecule. In contrast, singular potentials with N > 2 exhibit stronger divergences, and we argue why a complete theory cannot be recovered with finitely many parameters, drawing an analogy with nonrenormalizable theories in quantum field theory