On the Heat Kernel and Weyl Anomaly of Schr\"odinger invariant theory

Abstract

We propose a method inspired from discrete light cone quantization (DLCQ) to determine the heat kernel for a Schr\"odinger field theory (Galilean boost invariant with $z=2$ anisotropic scaling symmetry) living in $d+1$ dimensions, coupled to a curved Newton-Cartan background starting from a heat kernel of a relativistic conformal field theory ($z=1$) living in $d+2$ dimensions. We use this method to show the Schr\"odinger field theory of a complex scalar field cannot have any Weyl anomalies. To be precise, we show that the Weyl anomaly $\mathcal{A}^{G}_{d+1}$ for Schr\"odinger theory is related to the Weyl anomaly of a free relativistic scalar CFT $\mathcal{A}^{R}_{d+2}$ via $\mathcal{A}^{G}_{d+1}= 2\pi \delta (m) \mathcal{A}^{R}_{d+2}$ where $m$ is the charge of the scalar field under particle number symmetry. We provide further evidence of vanishing anomaly by evaluating Feynman diagrams in all orders of perturbation theory. We present an explicit calculation of the anomaly using a regulated Schr\"odinger operator, without using the null cone reduction technique. We generalise our method to show that a similar result holds for one time derivative theories with even $z>2$.