On the realization of impossible anomalies

The Wess-Zumino consistency condition allows more exotic forms of anomalies than those we usually encounter. For example in two-dimensional conformal field theories in the curved background with space-time dependent coupling constant $\lambda^i(x)$, a $U(1)$ current could possess anomalous divergence of the form $D^\mu J_\mu = \tilde{c} R + \chi_{ij} \partial^\mu \lambda^i \partial_\mu\lambda_j + \tilde{\chi}_{ij} \epsilon^{\mu\nu} \partial_\mu \lambda^i \partial_\nu \lambda^j + \cdots $. Another example is the CP odd Pontryagin density in four-dimensional Weyl anomaly. We could, however, argue that they are impossible in conformal field theories because they cannot correspond to any (unregularized) conformally invariant correlation functions. We find that this no-go argument may be a red herring. We show that some of these impossible anomalies avoid the no-go argument because they are not primary operators, and the others circumvent it because they are realized as semi-local terms as is the case with the conformally invariant Green-Schwartz mechanism and in the higher dimensional analogue of Liouville or linear dilaton theory.