Nonrelativistic inverse square potential, scale anomaly, and complex extension

TL;DR

This paper investigates the inverse square potential in quantum mechanics using a field-theoretic renormalization approach, exploring fixed points, limit cycles, and extending analysis to complex couplings to include inelastic scattering effects.

Contribution

It introduces a complex extension of the renormalization group analysis for the inverse square potential, providing analytical solutions and a geometric interpretation on the Riemann sphere.

Findings

Fixed points and limit cycles depend on the quadratic beta function discriminant.

Extension to complex couplings models inelastic scattering channels.

Analytical solutions of RG flow equations using bosonization.

Abstract

The old problem of a singular, inverse square potential in nonrelativistic quantum mechanics is treated employing a field-theoretic, functional renormalization method. An emergent contact coupling flows to a fixed point or develops a limit cycle depending on the discriminant of its quadratic beta function. We analyze the fixed points in both conformal and non-conformal phases and perform a natural extension of the renormalization group analysis to complex values of the contact coupling. Physical interpretation and motivation for this extension is the presence of an inelastic scattering channel in two-body collisions. We present a geometric description of the complex generalization by considering renormalization group flows on the Riemann sphere. Finally, using bosonization, we find an analytical solution of the extended renormalization group flow equations, constituting the main result of our work.