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I am a bit confused about Matthew D. Schwartz's statement of the Feynman rules in scalar QED (chapter 9, section 9.2 titled Feynman rules for scalar QED. The Lagrangian is \begin{equation} \mathcal{L} = -\frac{1}{4}F^2_{\mu\nu} -\phi^\star(\Box + m^2)\phi - i eA_\mu[\phi^\star(\partial_\mu\phi)-(\partial_\mu\phi^\star)\phi] + e^2 A^2_\mu|\phi|^2 \tag{9.11}. \end{equation} For clarity I am posting a screenshot of the equations that are concerned

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I don't understand how he writes that $\phi^\star(\partial_\mu\phi)$ annihilates the $e^{-}$ because I would think that the presence of $\phi$ in the other term $\phi(\partial_\mu\phi^\star)$ could also annihilate an electron. Can someone please explain?

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You are correct in saying that the $\phi$ in the term $\phi (\partial_{\mu} \phi^*)$ can also annihilate an electron, in fact otherwise the term $\phi (\partial_{\mu} \phi^*)$ would not contribute to the Feynman rule of the vertex.

The two term in the Lagrangian provide two separate contributions. For example consider $\phi^* (\partial_{\mu} \phi )$ alone. We have that

  1. $\phi^*$ must create the outgoing electron.
  2. $\phi$ must annihilate the incoming electron.
  3. $\partial_{\mu}$ acts on $\phi$ and spits out a term $-p^1_{\mu}$.

The term term $\phi (\partial_{\mu} \phi^*)$ will operate similarly and that is why the final Feynman rule has two different contributions.

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