Ghost-free completion of an effective matter coupling in bimetric theory

We consider a particular set of ghost-free interactions for three spin-2 fields in which we freeze out the dynamics of the metric tensor that couples to the matter sector. Integrating out the non-dynamical degrees of freedom in vacuum results in a ghost-free bimetric theory. In the presence of the matter source, which we treat as a small perturbation, the equations for the non-dynamical field can be solved perturbatively for its vierbein. This results in ghost-free bimetric theory in vierbein formulation with a modified matter coupling. To lowest order in matter perturbations, we precisely obtain an effective matter coupling that has been suggested earlier in the literature. This coupling contains a linear combination of the two vierbeine whose corresponding metric fluctuation coincides with the massless spin-2 mode. In the past, bimetric theory with this symmetric coupling has been treated as an effective theory valid at low energies without a ghost-free completion. Our results demonstrate that the effective matter coupling can be rendered entirely ghost-free by including the higher-order corrections obtained from the trimetric setup