The fate of conformal symmetry in the non-linear Schrödinger theory

The free Schrödinger theory in d   space dimensions is a non-relativistic conformal field theory. The interacting non-linear theory preserves this symmetry in specific numbers of dimensions at the classical (tree) level. This holds in particular for the |Φ|4|Φ|4-theory in d=2d=2. We compute the full quantum corrections to the 1PI 4-point function in d=2−ϵd=2−ϵ dimensions and find a non-trivial β  -function completely given by the 1-loop result. We exhibit an explicit Ward-identity showing that scale-invariance is broken in the limit d=2d=2 by an anomalous contribution proportional to the β-function.