  Recently, the possibility of inducing superconductivity for electrons in two
dimensional materials has been proposed via cavity-mediated pairing. The
cavity-mediated electron-electron interactions are long range, which has two
main effects: firstly, within the standard BCS-type pairing mediated by
adiabatic photons, the superconducting critical temperature depends
polynomially on the coupling strength, instead of the exponential dependence
characterizing the phonon-mediated pairing; secondly, as we show here, the
effect of photon fluctuations is significantly enhanced. These mediate novel
non-BCS-type pairing processes, via non-adiabatic photons, which are not
sensitive to the electron occupation but rather to the electron dispersion and
lifetime at the Fermi surface. Therefore, while the leading temperature
dependence of BCS pairing comes from the smoothening of the Fermi-Dirac
distribution, the temperature dependence of the fluctuation-induced pairing
comes from the electron lifetime. For realistic parameters, also including
cavity loss, this results into a critical temperature which can be more than
one order of magnitude larger than the BCS prediction. Moreover, a finite
average number photons (as can be achieved by incoherently pumping the cavity)
adds to the fluctuations and leads to a further enhancement of the critical
temperature.
