  The field theoretic renormalization study of reduced quantum electrodynamics
(QED) is performed up to two loops. In the condensed matter context, reduced
QED constitutes a very natural effective relativistic field theory describing
(planar) Dirac liquids, e.g., graphene and graphene-like materials, the surface
states of some topological insulators and possibly half-filled fractional
quantum Hall systems. From the field theory point of view, the model involves
an effective (reduced) gauge field propagating with a fractional power of the
d'Alembertian in marked contrast with usual QEDs. The use of the BPHZ
prescription allows for a simple and clear understanding of the structure of
the model. In particular, in relation with the ultra-relativistic limit of
graphene, we straightforwardly recover the results for both the interaction
correction to the optical conductivity: $\mathcal{C}^*=(92-9\pi^2)/(18\pi)$ and
the anomalous dimension of the fermion field: $\gamma_{\psi}(\bar{\alpha},\xi)
= 2 \bar{\alpha}\,(1-3\xi)/3 -16\,\left( \zeta_2 N_F + 4/27 \right)\,
\bar{\alpha}^2 + O(\bar{\alpha}^3)$, where $\bar{\alpha} = e^2/(4\pi)^2$ and
$\xi$ is the gauge-fixing parameter.
