Nonrelativistic scale anomaly, and composite operators with complex scaling dimensions

Research highlights: → Nonrelativistic scale anomaly leads to operators with complex scaling dimensions. → We study an operator O=ψψ in quantum mechanics with 1/r2 potenial. → The propagator of the composite operator is analytically computed. - 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=ψψ. We analytically compute the scaling dimension of this operator and determine the propagator <0|TOO+|0>. The operator O represents an infinite tower of bound states with a geometric energy spectrum. Operators with higher angular momenta are briefly discussed.