  Weyl semi-metals are three dimensional generalizations of graphene with
point-like Fermi surfaces. Their linear electronic dispersion leads to a window
in the particle-hole excitation spectrum which allows for undamped propagation
of collective excitations. We argue that interactions in Weyl semi-metals
generically lead to well-defined exciton modes. However, using a minimal model
for interactions, we show that the exciton binding energy is exponentially
small for weak interactions. This is due to effective two-dimensional character
in the space of particle-hole pairs that are available for bound state
formation. This is ultimately a consequence of linear electronic dispersion in
three dimensions. Nevertheless, intermediate interaction strengths can lead to
sharp spin-carrying excitonic resonances. We demonstrate this in a model Weyl
semi-metal with broken time-reversal symmetry and Hubbard interactions, using
GRPA (generalized random phase approximation) analysis. Excitons in Weyl
semi-metals have evoked interest as their condensation could lead to an axionic
charge density wave order. However, we find that the leading instability
corresponds to intra-valley spin density wave order which shifts the Weyl
points without opening a gap. Our results suggest interesting directions for
experimental studies of three dimensional Dirac systems.
