Lifshitz anomalies, Ward identities and split dimensional regularization

We analyze the structure of the stress-energy tensor correlation functions in Lifshitz field theories and construct the corresponding anomalous Ward identities. We develop a framework for calculating the anomaly coefficients that employs a split dimensional regularization and the pole residues. We demonstrate the procedure by calculating the free scalar Lifshitz scale anomalies in $2+1$ spacetime dimensions. We find that the analysis of the regularization dependent trivial terms requires a curved spacetime description without a foliation structure. We discuss potential ambiguities in Lifshitz scale anomaly definitions.