  Superconducting domes, ubiquitous across a variety of quantum materials, are
often understood as a window favorite for pairing opened by the fluctuations of
competing orders. Yet, a quantitative understanding of how such a window closes
is missing. Here, we show that inelastic neutron scattering, by quantifying a
length scale associated with polar fluctuations, $\ell_0$, addresses this
issue. We find that the superconducting dome of strontium titanate definitely
ends when $\ell_0$ vanishes. Moreover, the product of $\ell_0$ and the Fermi
wavevector peaks close to the maximum critical temperature. Thus, this
superconducting dome stems from the competition between the increase of the
density of states and the unavoidable collapse of the quantum paraelectric
phase, both induced by doping. The successful quantitative account of both the
peak and the end of the superconducting dome implies a central role in the
pairing mechanism played by the soft ferro-electric mode and its hybridisation
with the acoustic branch. Such a scenario may also be at work in other quantum
paraelectric materials, either bulk or interfaces.
