  About eight years ago it was predicted theoretically that a charged chiral
plasma could support the propagation of the so-called chiral magnetic waves,
which are driven by the anomalous chiral magnetic and chiral separation
effects. This prompted intensive experimental efforts in search of signatures
of such waves in relativistic heavy-ion collisions. In fact, several
experiments have already reported a tentative detection of the predicted
signal, albeit with a significant background contribution. Here, we critically
reanalyze the theoretical foundations for the existence of the chiral magnetic
waves. We find that the commonly used background-field approximation is not
sufficient for treating the waves in hot chiral plasmas in the long-wavelength
limit. Indeed, the back-reaction from dynamically induced electromagnetic
fields turns the chiral magnetic wave into a diffusive mode. While the
situation is slightly better in the strongly-coupled near-critical regime of
quark-gluon plasma created in heavy-ion collisions, the chiral magnetic wave is
still strongly overdamped due to the effects of electrical conductivity and
charge diffusion.
