  We extend the branch point twist field approach for the calculation of
entanglement entropies to time-dependent problems in 1+1-dimensional massive
quantum field theories. We focus on the simplest example: a mass quench in the
Ising field theory from initial mass $m_0$ to final mass $m$. The main
analytical results are obtained from a perturbative expansion of the twist
field one-point function in the post-quench quasi-particle basis. The expected
linear growth of the R\'enyi entropies at large times $mt\gg 1$ emerges from a
perturbative calculation at second order. We also show that the R\'enyi and von
Neumann entropies, in infinite volume, contain subleading oscillatory
contributions of frequency $2m$ and amplitude proportional to $(mt)^{-3/2}$.
The oscillatory terms are correctly predicted by an alternative perturbation
series, in the pre-quench quasi-particle basis, which we also discuss. A
comparison to lattice numerical calculations carried out on an Ising chain in
the scaling limit shows very good agreement with the quantum field theory
predictions. We also find evidence of clustering of twist field correlators
which implies that the entanglement entropies are proportional to the number of
subsystem boundary points.
