How to understand long-range propagate without decay in time and space? Assume there is a Green function:
$$G=\frac{1}{(p^2+r)-\sum-\omega^2}$$
where $\sum$ is self-energy. We know that if the self-energy vanishes, the quasi-particle is well-define, and it can propagate all the time without decay, i.e. lifetime is infinite. However, we also know that when $r>0$, the quasiparticle can only propagate in short-range, and decay in the correlation length $\xi\sim r^{-1/2}$.Only when $r=0$, such particle become massless and can propagate over long range, e.g. phonon.  What I am confused about is that for particles without self-energy, it has an infinite lifetime, why can't it still propagate long distances?
 A: The thing you are missing is relativity. A stable particle can propagate long distances - as long as it has enough time to do so without breaking the speed-of-light barrier.
The lifetime of a particle is proportional to $\mathrm{Im}\left[ \Sigma\left(-m^2 \right) \right]$, where $m$ is its mass, so it isn't actually necessary for the self-energy to completely vanish in order for the particle to be stable and have infinite lifetime; it's only necessary that $\Sigma(-m^2)$ be purely real. Nevertheless, for simplicity let's consider the simplest possible case of a stable particle - a free scalar field, where the self-energy does indeed vanish and
$$G({\bf p}, \omega) = \frac{1}{{\bf p}^2 - \omega^2 + m^2 }$$
(where my $m^2$ is your $r$).
Remember that the (real-space) propagator $G(x)$ is always a Lorentz-invariant function that only depends on the spacetime interval $s = x \cdot x$ of the argument $x$. For timelike $x$, the propagator is a Hankel function $-H(-s)/s = H(\tau^2)/\tau^2$, where $\tau$ is the proper time (I'm dropping lots of constants). For large $\tau$, this goes as $e^{i \tau^2}/\tau$, so the propagator decays very slowly (as a power law), which makes sense for a stable particle. For spacelike $x$, the propagator is a modified Bessel function $K_1(r)/r$ that decays rapidly (exponentially) as $e^{-r}/r^{3/2}$ for large $r$, where $r$ is the proper distance (not the coordinate distance).
So the point is that the particle can indeed propagate long distances, as long as it stays inside the light cone and has enough time to propagate over there "slower than the speed of light". What it can't do (except with vanishingly small probability) is propagate over large spacelike distances, as this would require "moving" faster than light.
