Nonrelativistic scale anomaly, and composite operators with complex scaling dimensions

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=ψψO=ψψ. We analytically compute the scaling dimension of this operator and determine the propagator 〈0|TOO†|0〉〈0|TOO†|0〉. The operator OO represents an infinite tower of bound states with a geometric energy spectrum. Operators with higher angular momenta are briefly discussed.

Research highlights
► Nonrelativistic scale anomaly leads to operators with complex scaling dimensions.
► We study an operator O=ψψO=ψψ in quantum mechanics with 1/r2 potenial.
► The propagator of the composite operator is analytically computed.