### 3.17 Comments on the entropy of interacting fields

So far we have considered free fields in a fixed gravitational (black hole) background. The interaction can
be included by adding a potential term to the classical action, here is
the set of fields in question. In the one-loop approximation, one splits , where is the
classical background field and is the quantum field. The integration over then reduces to the
calculation of the functional determinant of operator , where . The fields
representing the classical background are, in general, functions on the curved spacetime. In
some cases these fields are constants that minimize the potential . The matrix
plays the role of an -dependent mass matrix. In the approximation, when one can neglect
the derivatives of the matrix , the heat kernel of operator is presented as the
product . Using the already calculated trace of the heat kernel
on space with a conical singularity one obtains at one-loop the entanglement entropy of the
interacting fields. In dimensions one obtains [205] (see also [175] for a related discussion)
where we used that . Using the asymptotic behavior
, we find that the leading UV divergence of the entropy (152) is again (multiplied
by ) (81) and is thus not affected by the presence of the interaction in the action. However, the
interaction shows up in the sub-leading UV divergent and the UV finite terms. For instance, in four
dimensions on a flat background we find
We see that the leading UV divergent term proportional to the area is not modified by the presence of the
self-interaction. The mass matrix is a function of the background field . Thus, the
result (153) indicates that at tree-level the entropy should contain terms additional to those
of the standard area law, which depend on the value of field at the horizon. In order to
illustrate this point consider a model of a single field (a two-dimensional model of this type
was considered in [151], in four dimensions the role of self-interaction was discussed in [46])
where we included the term with the non-minimal coupling. In fact, if we had not included this term, it
would have been generated by the quantum corrections due to the self-interaction of the field . This is a
well-known fact, established in [182]. The renormalized non-minimal coupling in the model (154) is
where we omit the terms of higher order in . Splitting the field into classical and quantum parts in
Eq. (154) we find . Suppose for simplicity that the background metric is
flat. Then to leading order in the entanglement entropy (omitting the UV finite terms) is
This entropy should be considered as a quantum correction to the tree-level entropy
which follows from the action . We see that the logarithmic divergences in Eqs. (156) and
(155) agree if . On the other hand, the renormalization of (155) does not make the total entropy
completely UV finite. This is yet another manifestation of the puzzling behavior of
non-minimal coupling.