  The generalized hydrodynamic (GHD) approach has been extremely successful in
describing the out-of-equilibrium properties of a great variety of integrable
many-body quantum systems. It naturally extracts the large-scale dynamical
degrees of freedom of the system, and is thus a particularly good probe for
emergent phenomena. One such phenomenon is the presence of unstable particles,
traditionally seen via special analytic structures of the scattering matrix.
Because of their finite lifetime and energy threshold, these are especially
hard to study. In this paper we apply the GHD approach to a model possessing
both unstable excitations and quantum integrability. The largest family of
relativistic integrable quantum field theories known to have these features are
the homogeneous sine-Gordon models. We consider the simplest non-trivial
example of such theories and investigate the effect of an unstable excitation
on various physical quantities, both at equilibrium and in the non-equilibrium
state arising from the partitioning protocol. The hydrodynamic approach sheds
new light onto the physics of the unstable particle, going much beyond its
definition via the analytic structure of the scattering matrix, and clarifies
its effects both on the equilibrium and out-of-equilibrium properties of the
theory. Crucially, within this dynamical perspective, we identify unstable
particles as finitely-lived bound states of co-propagating stable particles of
different types, and observe how stable populations of unstable particles
emerge in large-temperature thermal baths.
