  Ultraviolet divergences are widely discussed in studies of entanglement
entropy. Also present, but much less understood, are infrared divergences due
to zero modes in the field theory. In this note, we discuss the importance of
carefully handling zero modes in entanglement entropy. We give an explicit
example for a chain of harmonic oscillators in 1D, where a mass regulator is
necessary to avoid an infrared divergence due to a zero mode. We also comment
on a surprising contribution of the zero mode to the UV-scaling of the
entanglement entropy.
