Large-$N_c$ constraints for elastic dark matter-light nucleus scattering in pionless effective field theory

Recent proposals for the use of light nuclei as dark matter direct detection targets necessitate a strong theoretical understanding of the nuclear physics involved. We perform relevant calculations for dark matter-light nucleus scattering in a combined pionless effective field theory and large-$N_c$ expansion, where $N_c$ is the number of quark colors. We include a general set of one-nucleon currents that have been used in other effective theories, as well as novel two-nucleon contact currents. First, we obtain constraints for the relative sizes of the dark matter couplings to the one- and two-nucleon currents through the large-$N_c$ expansion. Then, we use these constraints to make predictions for the relative sizes of spin-dependent and spin-independent cross sections for dark matter scattering off of a nucleon, a deuteron, a triton, and helium-3.