  We construct a low-energy effective field theory of electrons interacting via
short-range interactions in a Weyl semimetal and investigate possible
broken-symmetry ground states through an unbiased one-loop renormalization
group (RG) analysis. Using the symmetries of the noninteracting Weyl semimetal
to constrain the form of the interaction term leads to four independent
coupling constants. We investigate the stability of RG flows towards strong
coupling and find a single stable trajectory. In order to determine the most
likely broken-symmetry ground state, we calculate susceptibilities in the
particle-hole and particle-particle channels along this trajectory and find
that the leading instability is towards a fully gapped spin-density wave (SDW)
ground state. The sliding mode of this SDW couples to the external
electromagnetic fields like the Peccei-Quinn axion field of particle physics.
We also study a maximally symmetric toy model of an interacting Weyl semimetal
with a single independent coupling constant. The most likely ground states in
this case are either gapless ferromagnetic states where the spin waves couple
to the Weyl fermions like the spatial components of a (possibly chiral) gauge
field, or a fully gapped spin-singlet Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)
superconducting state.
