### 3.16 The puzzle of non-minimal coupling

The simplest case to consider is that of a

##### Non-minimally coupled scalar field.

In this case, one has , where is the parameter of
non-minimal coupling. The renormalization of Newton’s constant
is modified due to the presence of the non-minimal coupling in the scalar field operator. At the same
time, the entropy calculation on a Ricci flat background () is not affected by the non-minimal
coupling since the field operator for this background is identical to the minimal one. This simple
reasoning shows that the area law in the case of a non-minimally coupled scalar field is the
same as in the case of the minimal scalar field (81). Clearly, there is a mismatch between the
renormalization of Newton’s constant and the renormalization of the entanglement entropy. One concludes
that, in the presence of non-minimal coupling, when the Riemann tensor appears explicitly in
the action of the quantum field, the UV divergence of the entanglement entropy cannot be
handled by the standard renormalization of Newton’s constant. The mismatch in the entropy is
It is an important fact that there is no known way to give a statistical meaning to this entropy. Moreover,
(142) does not have a definite sign and may become negative if is positive. In some respects, this term is
similar to the classical Bekenstein–Hawking entropy: both entropies, at least in the framework of the
conventional field theory, do not have a well-defined statistical meaning. There is a hope that in string
theory the terms similar to Eq. (142) may acquire a better meaning. However, this question is still
open.
We should note that on a space with a conical singularity one can consider the Ricci scalar in the
non-minimal scalar operator as the complete curvature including the -like singular term as in Eq. (55).
Then, the differential operator contains a delta-like potential concentrated on the horizon
surface . The presence of this potential modifies the surface terms in the heat kernel in such a way
that [198]
where is the surface term (70) without the non-minimal coupling and the term
is ill defined (something like ). However, it does not affect the entropy calculation. If we
now apply the replica trick and calculate the entropy corresponding to the theory with the heat kernel with
the surface term (143) we get that [198, 163, 16]
This divergence takes a form consistent with the UV divergence of Newton’constant (141). However, we
cannot interpret this entropy as a contribution to the entanglement entropy since the presence of the
delta-like potential in the Euclidean field operator is not motivated from the point of view of
the original Lorentzian theory, for which the entanglement entropy is calculated. Moreover,
Eq. (144) is not positive if , while the entanglement entropy is supposed to be a positive
quantity.
Similar features are shared by other non–minimally-coupled fields.

##### Abelian vector field.

After gauge fixing, the partition function of an abelian gauge field is
where is operator defined in Eq. (132) with sign in the matrix (133). For the effective
action we find,
where
and is the number of on-shell degrees of freedom of the abelian vector field. The
renormalization of Newton’s constant is
Comparison with Eq. (126) shows that there is again a mismatch between the UV divergences in the
entropy and in Newton’s constant. This mismatch stems from the non-minimal term in the Laplace
type field operator for the vector field.

##### Massless graviton.

The partition function of a massless graviton in dimensions, after gauge fixing and
adding the Faddeev–Popov ghost contribution, is
where is operator defined in Eq. (132) with sign in the matrix (133). The
operator governs the dynamics of the tensor perturbations, which satisfy the condition
. The operator is due to the contribution of the conformal mode, while the
determinant of operator is due to the Faddeev–Popov ghosts. Hence, one has in this case that
where is the number of on-shell degrees of freedom of a massless spin- particle. The
renormalization of Newton’s constant is
Again, we observe the mismatch between the UV divergent terms in the entropy (126) and in Newton’s
constant, this time due to the graviton.
To summarize, the UV divergences in the entanglement entropy of minimally-coupled scalars and
fermions are properly renormalized by the redefinition of Newton’s constant. It happens that each
minimally-coupled field (no matter bosonic or fermionic) contributes positively to Newton’s constant and
positively to the entropy of the black hole. The contributions to both quantities come proportionally, which
allows the simultaneous renormalization of both quantities. The mismatch between the UV divergences in
the entropy and in Newton’s constant appears for gauge bosons: the abelian (and non-abelian)
vector fields and gravitons. The source of the mismatch are those non-minimal terms
in the field operator, which contribute negatively to Newton’s constant and do not make any
contribution to the entanglement entropy of the black hole. At the moment of writing of this
review the appropriate treatment of the entropy of non–minimally-coupled fields is not yet
available.