  Migdal's theorem plays a central role in the physics of electron-phonon
interactions in metals and semiconductors, and has been extensively studied
theoretically for parabolic band electronic systems in three-, two-, and
one-dimensional systems over the last fifty years. In the current work, we
theoretically study the relevance of Migdal's theorem in graphene and Weyl
semimetals which are examples of 2D and 3D Dirac materials, respectively, with
linear and chiral band dispersion. Our work also applies to 2D and 3D
topological insulator systems. In Fermi liquids, the renormalization of the
electron-phonon vertex scales as the ratio of sound ($v_s$) to Fermi ($v_F$)
velocity, which is typically a small quantity. In two- and three-dimensional
quasirelativistic systems, such as undoped graphene and Weyl semimetals, the
one loop electron-phonon vertex renormalization, which also scales as
$\eta=v_s/v_F$ as $\eta \rightarrow 0$, is, however, enhanced by an ultraviolet
\emph{logarithmic divergent correction}, arising from the linear, chiral Dirac
band dispersion. Such enhancement of the electron-phonon vertex can be
significantly softened due to the logarithmic increment of the Fermi velocity,
arising from the long range Coulomb interaction, and therefore, the
electron-phonon vertex correction does not have a logarithmic divergence at low
energy. Otherwise, the Coulomb interaction does not lead to any additional
renormalization of the electron-phonon vertex. Therefore, electron-phonon
vertex corrections in two- and three-dimensional Dirac fermionic systems scale
as $v_s/v^0_F$, where $v^0_F$ is the bare Fermi velocity, and small when $v_s
\ll v^0_F$. These results, although explicitly derived for the intrinsic
undoped systems, should hold even when the chemical potential is tuned away
from the Dirac points.
