  The nature of the gravitational interaction between ordinary and dark matter
is still open. Any deviation from universality or the Newtonian law also
modifies the standard assumption of collisionless dark matter. On the other
hand, obtaining a Yukawa-like large-distance modification of the gravitational
potential is a nontrivial problem, that has so far eluded a consistent
realization even at linearized level. We propose here a theory providing a
Yukawa-like potential, by coupling non-derivatively the two metric fields
related respectively to the visible and dark matter sectors, in the context of
massive gravity theories where the local Lorentz invariance is broken by the
different coexisting backgrounds. This gives rise to the appropriate mass
pattern in the gravitational sector, producing a healthy theory with the Yukawa
potential. Our results are of a special relevance in the scenario of dark
matter originated from the mirror world, an exact duplicate of the ordinary
particle sector.
