  We consider the parametric representation of the amplitudes of Abelian models
in the so-called framework of rank $d$ Tensorial Group Field Theory. These
models are called Abelian because their fields live on $U(1)^D$. We concentrate
on the case when these models are endowed with particular kinetic terms
involving a linear power in momenta. New dimensional regularization and
renormalization schemes are introduced for particular models in this class: a
rank 3 tensor model, an infinite tower of matrix models $\phi^{2n}$ over
$U(1)$, and a matrix model over $U(1)^2$. For all divergent amplitudes, we
identify a domain of meromorphicity in a strip determined by the real part of
the group dimension $D$. From this point, the ordinary subtraction program is
applied and leads to convergent and analytic renormalized integrals.
Furthermore, we identify and study in depth the Symanzik polynomials provided
by the parametric amplitudes of generic rank $d$ Abelian models. We find that
these polynomials do not satisfy the ordinary Tutte's rules
(contraction/deletion). By scrutinizing the "face"-structure of these
polynomials, we find a generalized polynomial which turns out to be stable only
under contraction.
