### 2.12 Entropy in theories with a modified propagator

In certain physically-interesting situations the propagator of a quantum field is different from the
standard and is described by some function as . The quantum field in question
then satisfies a modified Lorentz invariant field equation
Theories of this type naturally arise in models with extra dimensions. The deviations from the standard
form of propagator may be both in the UV regime (large values of ) or in the IR regime (small values of
). If the function for large values of grows faster than this theory is characterized by
improved UV behavior.
The calculation of the entanglement entropy performed in Section 2.9 can be generalized to include
theories with operator (33). This example is instructive since, in particular, it illuminates the exact relation
between the structure of 2-point function (the Green’s function in the case of free fields) and the
entanglement entropy [183].

In spacetime dimensions one has

Note that we consider Euclidean theory so that . The Green’s function
is a solution to the field equation with a delta-like source
and can be expressed in terms of the heat kernel as follows
Obviously, the Green’s function can be represented in terms of the Fourier transform in a manner similar to
Eq. (34),
The calculation of the trace of the heat kernel for operator (33) on a space with a conical singularity goes
along the same lines as in Section 2.9. This was performed in [184] and the result is
where the functions are defined as
The entanglement entropy takes the form (we remind the reader that for simplicity we take the surface
to be a -dimensional plane) [184]
It is important to note that [184, 183]
(i) the area law in the entanglement entropy is universal and is valid for any function ;

(ii) the entanglement entropy is UV divergent independently of the function , with the degree of
divergence depending on the particular function ;

(iii) in the coincidence limit, , the Green’s function (38)

may take a finite value if is decaying faster than . However, even for this function
, the entanglement entropy is UV divergent.
As an example, consider a function, which grows for large values of as . The 2-point
correlation function in this theory behaves as

and for it is regular in the coincidence limit. On the other hand, the entanglement entropy scales
as
and remains divergent for any positive value of . Comparison of Eqs. (43) and (44) shows that only for
(the standard form of the wave operator and the propagator) the short-distance behavior of the
2-point function is similar to the UV divergence of the entanglement entropy.