Renormalization and Particle Decay Recently I've been reading Sidney Coleman's QFT Notes (https://arxiv.org/abs/1110.5013) and there was one thing that I don't quite understand. 
In the notes (pg 102), he argues that a mass renormalization condition for a field $\phi$ which produces physical particles $|p\rangle$ is (I'm using relativistic normalization $\langle p'|p\rangle = (2\pi)^32E_p\delta^{(3)}(\vec{p}'-\vec{p})$)
$$\langle p'|S|p\rangle = (2\pi)^32E_p\delta^{(3)}(\vec{p}'-\vec{p})$$
i.e. that for the proper, renormalized, asymptotic states, there should be no scattering of a single particle since there is "nothing to scatter with" (the vacuum used is the physical vacuum $|\Omega\rangle$). The above equation should imply, I believe, that $S$ restricted to one particle states is the identity. I'm a bit confused because this seems to then imply that no decay is possible, but the Lagrangian he is working with includes an interaction term 
$$\mathcal{L}_I = -g\phi\psi^*\psi$$
which should allow for decay of a $\phi$ particle to a $\psi, \psi^*$ particle-antiparticle pair (say the mass $\mu$ of $\phi$ is greater than twice the mass of $\psi$). 
I assume that I'm missing something and Coleman is right. What am I missing?
 A: If there is a time decay $\tau_\alpha$ for a given particle it cannot be an element of the scattering states since this particle does not survive in $t\rightarrow\pm\infty$ limit. However, there are ways to work around this by taking $\tau_\alpha$ as very big, and not passing the limit $t\rightarrow\pm\infty$.
If the typical energies of the process $E$ are much larger than $1/\tau_\alpha$, i.e. $E\gg 1/\tau_\alpha$, we can use:
$$
d\Gamma(\alpha\rightarrow\beta)=2\pi |M_{\beta\alpha}|^2\delta^4(p_\beta-p_\alpha)\,d\beta
$$
as the decay rate of the process, where $M_{\beta\alpha}$ is defined as
$$
S_{\beta\alpha}=\delta(\beta-\alpha)-2\pi i\, M_{\beta\alpha} \delta^4(p_\beta-p_\alpha)
$$
and $S_{\beta\alpha}=\langle \beta|S|\alpha\rangle$ is the "fake" S-matrix where the time $t$ between the asymptotic past and future is not infinite but large. In this "fake" S-matrix the unstable particle does exist.
Now, the exact two-point function of scalar particles are given by:
$$
G^{(2)}(p)=\frac{i}{p^2-m_0-M^2(p^2)}
$$
if $M^2(p^2)$ have a non-zero imaginary part the behavior of the function around the pole is given by:
$$
G^{(2)}(p)\sim \frac{iZ}{p^2-m^2-iZ \text{Im} M^2(p^2)}
$$
where the time decay of the particle can be obtained as:
$$
\Gamma=-\frac{Z}{m}\text{Im}  M^2(m^2) = \frac{1}{m}\text{Im} \mathcal{M}(p\rightarrow p)
$$
where $\mathcal{M}(p\rightarrow p)$ is the $M_{\beta\alpha}$ for both $\beta$ and $\alpha$ being the unstable particle with momentum $p$. Now we need to search for convenient renormalization conditions to fix $Z$ and $m_0$. We cannot use the $\langle p'|S|p\rangle = (2\pi)^32E_p\delta^{(3)}(\vec{p}'-\vec{p})$ condition anymore, since there is a decay.
