  We demonstrate how virtual scattering of laser photons inside a cavity via
two-photon processes can induce controllable long-range electron interactions
in two-dimensional materials. We show that laser light that is
red(blue)-detuned from the cavity yields attractive(repulsive) interactions,
whose strength is proportional to the laser intensity. Furthermore, we find
that the interactions are not screened effectively except at very low
frequencies. For realistic cavity parameters, laser-induced heating of the
electrons by inelastic photon scattering is suppressed and coherent electron
interactions dominate. When the interactions are attractive, they cause an
instability in the Cooper channel at a temperature proportional to the square
root of the driving intensity. Our results provide a novel route for
engineering electron interactions in a wide range of two-dimensional materials
including AB-stacked bilayer graphene and the conducting interface between
LaAlO3 and SrTiO3.
