  An effective field theory (EFT) for dynamical axions in Weyl semimetals
(WSMs) is presented. A pseudoscalar axion excitation is predicted in WSMs at
sufficiently low temperatures, independently of the strength of the Weyl
fermion self-coupling. For strong fermion self-coupling the axion is the
gapless Goldstone boson of chiral $U(1)^{\text{ch}}$ spontaneous symmetry
breaking. For weak fermion self-coupling an axion is also generated at non-zero
chiral density for Weyl nodes displaced in energy, as a gapless collective mode
of correlated fermion pair excitations of the Fermi surface. This is an
explicit example of the extension of Goldstone's theorem to symmetry breaking
by the axial anomaly itself. In both cases the axion is a chiral density wave
or phason mode of the superfluid state of the WSM, and the Weyl fermions form a
chiral condensate $\langle\bar{\psi}\psi\rangle$ at low temperatures. In the
presence of an applied magnetic field the axion mode becomes gapped, in analogy
to the Anderson-Higgs mechanism in a superconductor. 't Hooft anomaly matching
from ultraviolet to infrared scales is directly verified in the EFT approach.
WSMs thus provide an interesting quantum system in which superfluid, non-Fermi
liquid behavior, and a dynamical axion are predicted to follow directly from
the axial anomaly in a consistent EFT that may be tested experimentally.
