  We study the dynamics of entanglement in the scaling limit of the Ising spin
chain in the presence of both a longitudinal and a transverse field. We present
analytical results for the quench of the longitudinal field in critical
transverse field which go beyond current lattice integrability techniques. We
test these results against a numerical simulation on the corresponding lattice
model finding extremely good agreement. We show that the presence of bound
states in the spectrum of the field theory leads to oscillations in the
entanglement entropy and suppresses its linear growth on the time scales
accessible to numerical simulations. For small quenches we determine exactly
these oscillatory contributions and demonstrate that their presence follows
from symmetry arguments. For the quench of the transverse field at zero
longitudinal field we prove that the R\'enyi entropies are exactly proportional
to the logarithm of the exponential of a time-dependent function, whose leading
large-time behaviour is linear, hence entanglement grows linearly. We conclude
that, in the scaling limit, linear growth and oscillations in the entanglement
entropies can not be simply seen as consequences of integrability and its
breaking respectively.
