  We study a nonlocal theory that combines both the Pseudo quantum
electrodynamics (PQED) and Chern-Simons actions among two-dimensional
electrons. In the static limit, we conclude that the competition of these two
interactions yields a Coulomb potential with a screened electric charge given
by $e^2/(1+\theta^2)$, where $\theta$ is the dimensionless Chern-Simons
parameter. This could be useful for describing the substrate interaction with
two-dimensional materials and the doping dependence of the dielectric constant
in graphene. In the dynamical limit, we calculate the effective current-current
action of the model considering Dirac electrons. We show that this resembles
the electromagnetic and statistical interactions, but with two different
overall constants, given by $e^2/(1+\theta^2)$ and $e^2\theta/(1+\theta^2)$.
Therefore, the $\theta$-parameter does not provide a topological mass for the
Gauge field in PQED, which is a relevant difference in comparison with quantum
electrodynamics. Thereafter, we apply the one-loop perturbation theory in our
model. Within this approach, we calculate the electron self-energy, the
electron renormalized mass, the corrected gauge-field propagator, and the
renormalized Fermi velocity for both high- and low-speed limits, using the
renormalization group. In particular, we obtain a maximum value of the
renormalized mass for $\theta\approx 0.36$. This behavior is an important
signature of the model and relations with doping control of band gap size are
also discussed in the conclusions.
