  We elucidate the mismatch between the $A$-anomaly coefficient and the
coefficient of the logarithmic term in the entanglement entropy of a Maxwell
field. In contrast to the usual assumptions about the protection of
renormalization group charges at the infrared, the logarithmic term is
different for a free Maxwell field and a Maxwell field interacting with heavy
charges. This is possible because of the presence of superselection sectors in
the IR theory. However, the correction due to the coupling with charged vacuum
fluctuations, that restores the anomaly coefficient, is independent of the
precise UV dynamics. The problem is invariant under electromagnetic duality,
and the solution requires both the existence of electric charges and magnetic
monopoles. We use a real-time operator approach but we also show how the
results for the free and interacting fields are translated into an effective
correction to the four-sphere partition function.
