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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?

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If there is a time decay $\tau_\alpha$, this particle cannot a entry of the S-matrix, i.e. this particle does not survive in $t\rightarrow\pm\infty$ limit. There are ways to work out this by doing $\tau_\alpha\sim t$, and not passing the limit $t\rightarrow\pm\infty$, then if the typical energies of the process $E$ are much greater 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. This is so because $\tau_\alpha$ is so big if we compare with $1/E$ that we can see this as $\tau_\alpha \sim t\gg1/E$. Here, $M_{\beta\alpha}$ 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 S-matrix.

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:

$$ G^{(2)}(p)\sim \frac{iZ}{p^2-m^2-iZ \Im M^2(p^2)} $$

where the time decay of the particle can be obtained as:

$$ \Gamma=-\frac{Z}{m}\Im M^2(m^2) = \frac{1}{m}\Im \mathcal{M}(p\rightarrow p) $$

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.

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