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For example, the Z boson decays and hence the propagator is: $$ \frac{1}{p^2-m^2-i \Gamma} $$ Where does this arise in QFT, is it that the lagrangian mass is complex or is it that when we compute the self-energy and get the renormalised mass we find that the propagator takes this form.

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The bubbles in the expansion of the$Z$ propagator become imaginary when the internal particles can go on-shell. The imaginary part is just the inverse lifetime of the $Z$.

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