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Consider, for example, the renormalized free field $\phi$-scalar Lagrangian (The question applies also to interacting $\phi^n$ theories, but that's not the issue):

$$\mathcal{L}=Z_2 \left(\partial_\mu\phi\right)^2 -Z_mm^2\phi^2=\mathcal{L}_0+\delta_2\left(\partial_\mu\phi\right)^2-\delta_m\phi^2$$

Where: $$\mathcal{L}_0=\left(\partial_\mu\phi\right)^2 -m^2\phi^2$$

The Feynman rules are:

Feynman rules for the free theory My question is why are they different in form? Why the original term $\sim\frac{i}{p^2}$, yet the counter term $\sim p^2$.

Remark: The mass term can be disregraded in the answer, since I've established that by taking the mass as an interaction to a free-massless theory, we can take the massive-interaction $\sigma(p^2)$ as:

$$\sigma(p^2)=\frac{i}{p^2}+\frac{i}{p^2}\left(-im^2\right)\frac{i}{p^2}+\frac{i}{p^2}\left(-im^2\right)\frac{i}{p^2}\left(-im^2\right)\frac{i}{p^2}\dots=\frac{i}{p^2-m^2}$$

Thank you in advance.

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marked as duplicate by AccidentalFourierTransform, Community Aug 18 '17 at 11:47

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