We present an effective field theory for interacting electrons in clean semimetals (both three dimensional Dirac semimetals and graphene) in terms of their soft or massless bosonic degrees of freedom. We show, by means of a Ward identity, that the intrinsic semimetal ground state breaks the Sp(4M) symmetry of the theory. In Fermi liquids this enables one to identify the massive, non-Goldstone modes of the theory and integrate them out. Due to the vanishing density of states in semimetals, unlike in Fermi liquids, both Goldstone and non-Goldstone modes are equally soft, and so all two-particle correlations need to be kept. The resulting theory is not perturbative with respect to the electron-electron interaction; rather, it is controlled by means of a systematic loop expansion and allows one to determine the exact asymptotic form of observables in the limits of small frequencies and/or wave vectors. Equivalently, it provides a mechanism of determining the long time-tail and long wavelength behavior of observables and excitations. As a representative application, we use the theory to compute the zero-bias anomaly for the density of states for both short and long-range interactions in two and three dimensions. We find that the leading nonanalyticity in semimetals with a long-ranged interaction appears at the same order in frequency as the one in Fermi liquids, since the effects of the vanishing density of states at the Fermi level are offset by the breakdown of screening. Consequently, we are able to provide a logical scheme to determine the leading non-analytical behavior of observables in semimetals using knowledge of the corresponding non-analyticities in a Fermi liquid. / 2020-01-11