Resonance propagator properties

To start, we would like to calculate polarization operator (=fermion bubble) in scalar Yukawa theory, $gbar{psi}phipsi$. The expression for polaziation operator is $$Pi(k^2)=(-1)(-g)^2int_pfrac{(gammacdot p+m)(gamma(p+k)+m))}{[(p^2-m^2)-iepsilon][(p+k)^2-m^2+iepsilon]}$$ Then, you can show that it is possible to sum the series with bubbles (it is just geometric series). The key point is to understand when $Pi(k^2)$ obtains imaginary part. You can show it in a variety of ways (I can write down the calculation). The main thesis is

Polarization operator has imaganiry part if $k^2>4m^2$, where $m$ is fermion mass in the theory and $k$ is scalar 4-momentum

I assume that if you perform accurate calculation of $Pi(k^2)$ and then consider renormalized propagator, you should find something like (in one bubble approximation) $$frac{i}{k^2-M^2-Pi(k^2)}$$ and if $Pi(k^2)$ has imagianary part you exactly obtain your formula.

Indeed, one of the possible ways to calculate $Pi(k^2)$ is to cut diagram and consider decay of scalar particle. Decay width will be $$Gamma=frac{g^2}{8pi M}k^2sqrt{1-frac{4m^2}{k^2}}$$ and it relates to imaginary part as $text{Im},Pi(k^2)=-MGamma$. To see it, you can just calculate $text{Im},Pi(k^2)$ using another way and compare results.

If my answer is unclear or you need to see all the derivations, I can provide derivations and try to clarify my statements.

Suggested strategy is:

1. Consider 1-loop polarization operator and show how it obtains imaginary part
2. Then consider renormalized scalar propagator
3. Understand how to sum all 1PI (=1-particle irreducible) diagrams

So this is what I have until now. For the normal propagator (no loop) we get: $$frac{i}{p^2-M^2+iepsilon}$$ M is the mass of $phi$ and m is the mass of $psi$. For the 1 loop correction, I get: $$(frac{i}{p^2-M^2+iepsilon})^2int d^4kfrac{i}{k^2-m^2+iepsilon}frac{i}{(p-k)^2-m^2+iepsilon}$$ For 2 loops $$(frac{i}{p^2-M^2+iepsilon})^3(int d^4kfrac{i}{k^2-m^2+iepsilon}frac{i}{(p-k)^2-m^2+iepsilon})^2$$ so the series will look in the end like this: $$frac{i}{p^2-M^2+iepsilon}frac{1}{1-frac{i}{p^2-M^2+iepsilon}int d^4kfrac{i}{k^2-m^2+iepsilon}frac{i}{(p-k)^2-m^2+iepsilon}}$$ $$frac{i}{p^2-M^2+iepsilon-iint d^4kfrac{i}{k^2-m^2+iepsilon}frac{i}{(p-k)^2-m^2+iepsilon}}$$ I assume we can get rid of the first $iepsilon$ term as we don't use it for any integral so we get $$frac{i}{p^2-M^2+iint d^4kfrac{1}{k^2-m^2+iepsilon}frac{1}{(p-k)^2-m^2+iepsilon}}$$ and I think I need to show that this is equal to $$frac{i}{p^2-M^2+i M Gamma}$$ but I am not sure how. Also I am not sure what do they mean by "Show that if these give an imaginary number..." Do I need to assume that the integral is an imaginary number? And what then? Can someone tell me if this is correct and help me from here please?