Lifshitz electrodynamics with Luttinger fermions

Three-dimensional Luttinger fermions have quadratic band touching dispersion with Lifshitz-like dynamical scaling z = 2. We extend the single-particle Luttinger Hamiltonian to include the effects of minimally coupled gauge fluctuations and illustrate the long-wavelength behavior of the combined interacting system. Within the parameter space of our model, we discover an analog of the non-Fermi liquid phase found in Luttinger semimetals with static Coulomb interaction. We further show that dynamical gauge fluctuations can also lead to non-relativistic Lifshitz scaling for the emergent photons in our theory. With an RG analysis, we describe the stability and scaling properties of its fixed points in the presence of additional cubic-symmetry allowed perturbations.