  We consider a system of interacting fermions in two dimensions beyond the
second-order perturbation theory in the interaction. It is shown that the
mass-shell singularities in the self-energy, arising already at the second
order of the perturbation theory, manifest a non-perturbative effect: an
interaction with the zero-sound mode. Resumming the perturbation theory for a
weak, short-range interaction and accounting for a finite curvature of the
fermion spectrum, we eliminate the singularities and obtain the results for the
quasi-particle self-energy and the spectral function to all orders in the
interaction with the zero-sound mode. A threshold for emission of zero-sound
waves leads a non-monotonic variation of the self-energy with energy (or
momentum) near the mass shell. Consequently, the spectral function has a
kink-like feature. We also study in detail a non-analytic temperature
dependence of the specific heat, $C(T)\propto T^2$. It turns out that although
the interaction with the collective mode results in an enhancement of the
fermion self-energy, this interaction does not affect the non-analytic term in
$C(T)$ due to a subtle cancellation between the contributions from the real and
imaginary parts of the self-energy. For a short-range and weak interaction,
this implies that the second-order perturbation theory suffices to determine
the non-analytic part of $C(T)$. We also obtain a general form of the
non-analytic term in $C(T)$, valid for the case of a generic Fermi liquid,
\emph{i.e.}, beyond the perturbation theory.
