  The large-scale dynamics of quantum integrable systems is often dominated by
ballistic modes due to the existence of stable quasi-particles. We here
consider as an archetypical example for such a system the spin-$\frac{1}{2}$
XXX Heisenberg chain that features magnons and their bound states. An
interesting question, which we here investigate numerically, arises with
respect to the fate of ballistic modes at finite temperatures in the limit of
zero magnetization $m{=}0$. At a finite magnetization density $m$, the spin
autocorrelation function $\Pi(x,t)$ (at high temperatures) typically exhibits a
trimodal behavior with left- and right-moving quasi-particle modes and a broad
center peak with slower dynamics. The broadening of the fastest propagating
modes exhibits a sub-diffusive $t^{1/3}$ scaling at large magnetization
densities, $m {\rightarrow} \frac{1}{2}$, familiar from non-interacting models;
it crosses over into a diffusive scaling $t^{1/2}$ upon decreasing the
magnetization to smaller values. The behavior of the center peak appears to
exhibit a crossover from transient super-diffusion to ballistic relaxation at
long times. In the limit $m{\to}0$, the weight carried by the propagating peaks
tends to zero; the residual dynamics is carried only by the central peak; it is
sub-ballistic and characterized by a dynamical exponent $z$ close to the value
$\frac{3}{2}$ familiar from Kardar-Parisi-Zhang (KPZ) scaling. We confirm,
employing elaborate finite-time extrapolations, that the spatial scaling of the
correlator $\Pi$ is in excellent agreement with KPZ-type behavior and analyze
the corresponding corrections.
