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

Branch Point Twist Field Form Factors in the sine-Gordon Model I: Breather Fusion and Entanglement Dynamics