Mass dimension one fermions and their gravitational interaction.

We investigate in detail the interaction between the spin-${1/2}$ fields endowed with mass dimension one and the graviton, up to the first order of approximation in ${\kappa}$. We obtain an interaction vertex that combines the characteristics of scalar-graviton and Dirac's fermion-graviton vertices, due to the scalar-dynamic attribute and the fermionic structure of this field. It is shown that the founded vertex obeys the Ward-Takahashi identity, ensuring the gauge invariance for this interaction. In the contribution of the mass dimension one fermion to the graviton propagator at one-loop, we found that the conditions for the tadpole term can be canceled by a cosmological counter-term in a higher-derivative theory for gravity in the covariant formalism. Finally, we calculate the scattering process at the non-relativistic limit, describing the newtonian potential mediated by the graviton. The result reveals that only the scalar sector present in the vertex contributes to the gravitational potential, since the character of Fermi-Dirac statistics that would arise due to the fermionic term, with their respective helicities, does not manifest in the non-relativistic limit.