  We study the entanglement entropy of a region of length 2L with the remainder
of an infinite one dimensional gapless quantum system in the case where the
region is centered on a quantum impurity. The coupling to this impurity is not
scale invariant, and the physics involves a crossover between weak and strong
coupling regimes. While the impurity contribution to the entanglement has been
computed numerically in the past, little is known analytically about it, since
in particular the methods of conformal invariance cannot be applied because of
the presence of a crossover length. We show in this paper that the small
coupling expansion of the entanglement entropy in this problem is quite
generally plagued by strong infrared divergences, implying a non-perturbative
dependence on the coupling. The large coupling expansion turns out to be better
behaved, thanks to powerful results from the boundary CFT formulation and, in
some cases, the underlying integrability of the problem. However, it is clear
that this expansion does not capture well the crossover physics. In the
integrable case -- which includes problems such as an XXZ chain with a modified
link, the interacting resonant level model or the anisotropic Kondo model -- a
non perturbative approach is in principle possible using form-factors. We adapt
in this paper the ideas of [1,2] to the gapless case and show that, in the
rather simple case of the resonant level model, and after some additional
renormalizations, the form factors approach yields remarkably accurate results
for the entanglement all the way from short to large distances. This is
confirmed by detailed comparison with numerical simulations. Both our form
factor and numerical results are compatible with a non-perturbative form at
short distance.
