  The generation of a chiral shift parameter in the normal ground state of
magnetized relativistic matter is discussed. The chiral shift contributes to
the axial current density, but does not modify the conventional axial anomaly
relation. The analysis based on gauge invariant regularization schemes suggests
that this finding is also valid in gauge theories. We argue that the chiral
shift parameter can affect observable properties of compact stars. In the
regime of heavy ion collisions, the chiral shift parameter can modify the key
features of the chiral magnetic effect.
