  In the context of quantum field theory (QFT), unstable particles are
associated with complex-valued poles of two-body scattering matrices in the
unphysical sheet of rapidity space. The Breit-Wigner formula relates this pole
to the mass and life-time of the particle, observed in scattering events. In
this letter, we uncover new, dynamical signatures of unstable excitations and
show that they have a strong effect on the non-equilibrium properties of QFT.
Focusing on a 1+1D integrable model, and using the theory of Generalized
Hydrodynamics, we study the formation and decay of unstable particles by
analysing the release of hot matter into a low-temperature environment. We
observe the formation of tails and the decay of the emitted nonlinear waves, in
sharp contrast to the situation without unstable excitations. We expect these
signatures of instability to have a large degree of universality. Our study
shows that the out-of-equilibrium dynamics of many-body systems can be strongly
affected not only by the spectrum, but also by excitations with finite
life-times.
