After a short introduction on the delicate and crucial treatment of chiral anomaly in axion physics we will discuss a current controversy on the trace anomaly. The existence of CP violating terms in the trace of the energy-momentum tensor (EMT) in chiral theories coupled to gravity has no answer yet. Consequently, the neutrino could carry a source of CP or unitarity violation in its trace anomaly within the Standard Model. This black board talk will aim to clarify the situation by presenting two exclusive non-perturbative computations of the trace anomaly. The first approach will deal with path integral methods to obtain directly the trace anomaly of a Weyl fermion. The second approach will be more generic since it will proceed from the EMT relying only on symmetry arguments.