  We extend the entanglement entropy calculation performed in the seminal paper
by Srednicki for free real massive scalar field theories in 1+1, 2+1 and 3+1
dimensions. We show that the inverse of the scalar field mass can be used as an
expansion parameter for a perturbative calculation of the entanglement entropy.
We perform the calculation for the ground state of the system and for a
spherical entangling surface at third order in this expansion. The calculated
entanglement entropy contains a leading area law term, as well as subleading
terms that depend on the regularization scheme, as expected. Universal terms
are non-perturbative effects in this approach. Interestingly, this perturbative
expansion can be used to approximate the coefficient of the area law term, even
in the case of a massless scalar field in 2+1 and 3+1 dimensions. The presented
method provides the spectrum of the reduced density matrix as an intermediate
result, which is an important advantage in comparison to the replica trick
approach. Our perturbative expansion underlines the relation between the area
law and the locality of the underlying field theory.
