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- One-loop correction for $\phi \to e^+e^-$ under a Yukawa interaction seems to vanish trivially.

I am trying to calculate the amplitude for a decay ##\phi \to e^+e^-## under a Yukawa interaction ##\mathcal{L}_I = -g\phi \bar{\psi}\psi## to one-loop order (with massless fermions for simplicity).

If I'm not wrong, there are 4 diagrams that contribute to 1 loop, three diagrams involving self-energy corrections (i.e. inserting a loop into the external lines) and an extra diagram with vertex correction (a ##\phi## field exchanged by ##e^+## and ##e^-##).

I have no problem calculating the integrals and using counterterms to cancel the infinities that arise, but I'm not sure if the conditions I use for renormalization are correct. Following the example of QED, to apply on-shell renormalization I used the following conditions;

The scalar propagator in the limit ##p^2 \to M^2## should be ##\frac{i}{p^2-M^2}##

The fermion propagator in the limit ##\not{\!p} \to 0## should be ##\frac{i}{\not{p}}##

The vertex function in the limit ##p^2 \to M^2## should be ##-ig##. (##p## is the momentum of the scalar particle.)

Now, because the self-energy diagrams are all in external legs, the first two corrections mean that those diagrams vanish.

But the third condition tells that the vertex correction must also vanish when the scalar particle is on-shell (as in my diagram). Therefore all the diagrams here vanish trivially due to renormalization conditions.

Is this analysis correct? Or did I make some mistake in the renormalization part?

If I'm not wrong, there are 4 diagrams that contribute to 1 loop, three diagrams involving self-energy corrections (i.e. inserting a loop into the external lines) and an extra diagram with vertex correction (a ##\phi## field exchanged by ##e^+## and ##e^-##).

I have no problem calculating the integrals and using counterterms to cancel the infinities that arise, but I'm not sure if the conditions I use for renormalization are correct. Following the example of QED, to apply on-shell renormalization I used the following conditions;

The scalar propagator in the limit ##p^2 \to M^2## should be ##\frac{i}{p^2-M^2}##

The fermion propagator in the limit ##\not{\!p} \to 0## should be ##\frac{i}{\not{p}}##

The vertex function in the limit ##p^2 \to M^2## should be ##-ig##. (##p## is the momentum of the scalar particle.)

Now, because the self-energy diagrams are all in external legs, the first two corrections mean that those diagrams vanish.

But the third condition tells that the vertex correction must also vanish when the scalar particle is on-shell (as in my diagram). Therefore all the diagrams here vanish trivially due to renormalization conditions.

Is this analysis correct? Or did I make some mistake in the renormalization part?

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