  We investigate the possibilities of searching for a self-conjugate
polarizable particle in the self-interactions of light. We first observe that
polarizability can arise either from the exchange of mediator states or as a
consequence of the inner structure of the particle. To exemplify this second
possibility we calculate the polarizability of a neutral bosonic open string,
and find it is described only by dimension-8 operators. Focussing on the spin-0
case, we calculate the light-by-light scattering amplitudes induced by the
dimension-6 and 8 polarizability operators. Performing a simulation of
exclusive diphoton production with proton tagging at the LHC, we find that the
imprint of the polarizable dark particle can be potentially detected at
5$\sigma$ significance for mass and cutoff reaching values above the TeV scale,
for $\sqrt{s}=$13~TeV and 300 fb$^{-1}$ of integrated luminosity. If the
polarizable dark particle is stable, it can be a dark matter candidate, in
which case we argue this exclusive diphoton search may complement the existing
LHC searches for polarizable dark matter.
