  Describing experimental signatures of quantum spin ice has been the focus of
many theoretical efforts, as definitive experimental verification of this
candidate quantum spin liquid is yet to be achieved. Gapped excitations known
as visons have largely eluded those efforts. We provide a theoretical
framework, which captures their dynamics and predicts new signatures in the
magnetic response. We achieve this by studying the ring-exchange Hamiltonian of
quantum spin ice in the large-$s$ approximation, taking into account the
compact nature of the emergent $U(1)$ gauge theory. We find the stationary
solutions of the action -- the instantons -- which correspond to visons
tunneling between lattice sites. By integrating out the instantons, we
calculate the effective vison Hamiltonian, including their mass. We show that
in the ground state virtual vison pairs simply renormalise the speed of light.
At low temperatures, however, thermally activated visons form a Debye plasma
and introduce a mass gap in the photon spectrum, equal to the plasma frequency,
which we calculate as a function of temperature. We demonstrate that this
dynamical mass gap should be visible in energy-resolved neutron scattering
spectra but not in the energy-integrated ones. We also show that it does not
lead to confinement of static spinons.
