  We present a comparative analysis of the validity of Eliashberg theory for
the cases of fermions interacting with an Einstein phonon and with soft nematic
fluctuations near an Ising-nematic/Ising-ferromagnetic quantum-critical point
(QCP). In both cases, Eliashberg theory is obtained by neglecting vertex
corrections. For the phonon case, the reasoning to neglect vertex corrections
is the Migdal ``fast electron/slow boson'' argument because the phonon velocity
is much smaller than the Fermi velocity, $v_F$. The same argument allows one to
compute the fermionic self-energy within Eliashberg theory perturbatively
rather than self-consistently. For the nematic case, the velocity of a
collective boson is comparable to $v_F$ and this argument apparently does not
work. Nonetheless, we argue that while two-loop vertex corrections near a
nematic QCP are not small parametrically, they are small numerically. At the
same time, perturbative calculation of the fermionic self-energy can be
rigorously justified when the fermion-boson coupling is small compared to the
Fermi energy. Furthermore, we argue that for the electron-phonon case
Eliashberg theory breaks down at some distance from where the dressed Debye
frequency would vanish, while for the nematic case it holds all the way to a
QCP. From this perspective, Eliashberg theory for the nematic case actually
works better than for the electron-phonon case.
