  We construct, from first principles, a covariant local model for scalar
fractonic matter coupled to a symmetric tensor gauge field. The free gauge
field action is just the one of Blasi-Maggiore model. The scalar sector is a
non-trivial covariant generalization of Pretko's quartic model. Because the
model has no quadratic term in the scalar field, a direct perturbative
treatment fails. Remarkably, we demonstrate that the action can be driven to a
perturbative effective action. However, with the price of carrying non-local
interacting terms. We study the perturbative regime of the model and show that
there are at least four distinct phases in the model: One with massive
fractonic modes; one with massless fractons; a mixed phase with massive and
massless fractons; and another one where fractons cannot occur at all in the
physical spectrum.
