  The quantum sine-Gordon model is the simplest massive interacting integrable
quantum field theory whose two-particle scattering matrix is generally
non-diagonal. As such, it is a model that has been extensively studied,
especially in the context of the bootstrap programme. In this paper we compute
the form factors of a special local field known as the branch point twist
field, whose correlation functions are building blocks for measures of
entanglement. We consider the attractive regime where the theory posesses a
particle spectrum consisting of a soliton, an antisoliton (of opposite $U(1)$
charges) and several (neutral) breathers. In the breather sector we exploit the
fusion procedure to compute form factors of heavier breathers from those of
lighter ones. We apply our results to the study of the entanglement dynamics
after a small mass quench and for short times. We show that in the presence of
two or more breathers the von Neumann and R\'enyi entropies display undamped
oscillations in time, whose frequencies are proportional to the even breather
masses and whose amplitudes are proportional to the breather's one-particle
form factor.
