  It is generally accepted that the phonon gas of a superfluid always enters a
weak coupling regime at sufficiently low temperatures, whatever the strength of
the interactions between the underlying particles (constitutive of the
superfluid). Thus, in this limit, we should always be able to calculate the
damping rate of thermal phonons by applying Fermi's golden rule to the
Hamiltonian $H_3$ of cubic phonon-phonon coupling taken from quantum
hydrodynamics, at least in the case of a convex acoustic branch and in the
collisionless regime (where the eigenfrequency of the considered phonons
remains much greater than the gas thermalization rate). Using the many-body
Green's function method, we predict that, unexpectedly, this is not true in two
dimensions, contrary to the three-dimensional case. We confirm this prediction
with classical phonon-field simulations and a non-perturbative theory in $H_3$,
where the fourth order is regularized by hand, giving a complex energy to the
virtual phonons of the four-phonon collisional processes. For a weakly
interacting fluid and a phonon mode in the long-wavelength limit, we predict a
damping rate about three times lower than that of the golden rule.
