Nonrelativistic scale anomaly, and composite operators with complex scaling dimensions

TL;DR

This paper demonstrates how a nonrelativistic quantum scale anomaly leads to composite operators with complex scaling dimensions, revealing an infinite tower of bound states with a geometric energy spectrum.

Contribution

It analytically computes the scaling dimension of a composite operator in nonrelativistic quantum mechanics with an inverse square potential, highlighting the emergence of complex dimensions due to the anomaly.

Findings

Operators with complex scaling dimensions indicate a scale anomaly.

The composite operator O creates an infinite tower of bound states.

The energy spectrum of bound states is geometric.

Abstract

It is demonstrated that a nonrelativistic quantum scale anomaly manifests itself in the appearance of composite operators with complex scaling dimensions. In particular, we study nonrelativistic quantum mechanics with an inverse square potential and consider a composite s-wave operator O=\psi\psi. We analytically compute the scaling dimension of this operator and determine the propagator <0|T O O^{\dagger}|0>. The operator O represents an infinite tower of bound states with a geometric energy spectrum. Operators with higher angular momenta are briefly discussed.