How to understand complex masses of unstable particles? The conceptual problem of calculating decay rate If a particle has a complex mass, $p^2-m^2=0$ leads to $p^μ \notin \mathbb R^4$. What does it mean?
When you want to calculate S-matrix elements of decay process $\langle p_f,\ldots\mid p_i\rangle$, you compute the $n$-point correlation function $$\langle 0\mid Tφ(x_1)φ(x_2)\ldots\mid 0\rangle$$ and using the LSZ formula. Fourier transformation of the correlation function has the poles at $p_j^2=m_j^2$ ,because it includes the vacuum polarization amplitudes$$ = \frac i {p_j^2-m_j^2+O((p_j^2-m_j^2)^2)}\approx \frac i{p_i^2-m_i^2}$$ on each external legs in diagrams. The LSZ formula says when you multiply $Π_j(p_j^2-m_j^2)/i$ and take the  limit $p_j^2→m_j^2$, you get S-matrix elements $\langle p_f,\ldots\mid p_i\rangle$. Therefore, you may know that you need to consider only "amputated diagrams" when you calculate S-matrix.
However, beyond the tree level, the vacuum polarization amplitude of unstable particle has the pole at not real but complex value. This means on shell limit $p_i^2→m_i^2$ leads to $p_i^\mu \notin \mathbb R^4$. On the other hand, if you take limit such as $p_i^2\to\operatorname{Re} m_i^2$, you cannot do the approximation "vacuum polarization amplitudes $ \approx \frac i{p_i^2-m_i^2}$", because $p_i^2$ is far from the pole.
After all, my question is how to compute decay rates of unstable particles correctly. Can I calculate S-matrix elements by computing "amputated diagrams" as usually done?
The related problem is noted in Srednicki's textbook on P.162
http://web.physics.ucsb.edu/~mark/qft.html
He said in and out state should consist of infinitely long-lived particles. He thought unstable particles present as intermediate states, and regarded decay late as a quantity related to the width of resonance. (p.165 (25.25))
$$f(E)=\frac{1}{E-E_0+iΓ/2}.$$
Fourier transformation of $f(E)$ gives $g(t)=\exp(iE_0t-Γt/2)$, which seems something expressing decay. I want to know too the precise meaning of this procedure. (At first, I considered the time evolution of $|ψ\rangle = \int dE \, f(E)|E\rangle$ by Schrodinger equation, but $|ψ(t)\rangle\neq g(t)|ψ(0)\rangle$).
 A: I'll discuss this in the context of Yukawa theory, and use renormalized perturbation theory. The way I understand it is this. 
Setup: Consider the renormalization scheme 
$$ \psi_R = \frac{1}{\sqrt{Z}}\psi_0 \\
m_R = \frac{1}{Z_m} m_0 $$
where $Z_i = 1 + \delta_i$. Then the renormalized propagator is.
$$G^R = \frac{1}{Z_2} G^{bare}. $$
Now recall that summing over all 1PI insertions we may write 
$$iG^{bare} = \frac{i}{\not{p} - m_0 + \Sigma(\not{p})}$$
where $\Sigma$ is the sum over all 1PI insertions. Now, we can write $\Sigma = \Sigma_2 + {\cal O}(g^4)$ keeping only those insertions of order less than $g^4$. So that our renormalized propagator to order $g^2$ is
$$G^R =  \frac{1}{1+\delta}\frac{i}{\not{p} - m_0 + \Sigma_2(\not{p})} \\
\boxed{G^R= \frac{i}{\not{p} - m_R + \Sigma_R(\not{p})} } $$
where we have defined 
$$ 
\Sigma_R = \Sigma_2 + \delta \not{p} - (\delta + \delta_m)m_R 
$$
Now, recall that we define the physical mass as the pole of the propagator. Therefore, 
$$ m_P - m_R + \Sigma_R(m_P) = 0 .$$
This implies that 
$$ \boxed{\delta_m m_P = \Sigma_R(m_P)} $$

Your question: Let us call $\Sigma_2(\not{p}) = i\Delta m$, anticipating that the correction is purely imaginary. Then, our propagator is 
$$\tilde{G}^R= \frac{i}{\not{p} - m_R+ i\Delta m} $$
where now we know that our propagator has a pole at $m_P$. 
Recall that the amplitude for a particle to propagate from $x$ to $y$ is given by the fourier transform 
$$D(x-y) = \int \frac{d^4p}{(2\pi)^4} \frac{i}{\not{p} - m_R + i\Delta m}e^{-ip(x-y)} $$
To make things easier suppose that y=0. Then we have 
$$D(x) = \int \frac{d^4p}{(2\pi)^4} \frac{i}{\not{p} - m_R+ i\Delta m}e^{i\mathbf{p\cdot x}}e^{-iEt} $$
Where $E = \sqrt{p^2 + m_{rest}^2}$. What is the mass of the particle in its rest frame? Well by definition it's the pole of the propagator! And so we may replace 
$$E = \sqrt{p^2 + (m_R - i \Delta m)^2} \\
=\sqrt[4]{\left(-{\Delta m}^2+m^2+p^2\right)^2+4 {\Delta m}^2 m^2} \left(i \sin \left(\frac{\phi}{2}\right)+\cos \left(\frac{\phi}{2}\right)\right)\\
\boxed{E \equiv \xi \left(i \sin \left(\frac{\phi}{2}\right)+\cos \left(\frac{\phi}{2}\right)\right) }$$
where $\phi$ is the $\mathrm{Arg}$ of the radicand.Finally, we obtain that 
$$ \boxed{D(x) = \int \frac{d^4p}{(2\pi)^4} \frac{ie^{i\mathbf{p\cdot x}} e^{-i\xi\cos(\frac{\phi}{2}) t}}{\not{p} - m_R+ i\Delta m}e^{-\xi\sin(\frac{\phi}{2}) t}}  $$
and so we see that indeed the probability for a particle to exist at a time $t$ decays exponentially. 
