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I am trying to study QCD renormalization at 1-loop order. So, when I take into account the gluon self-energy corrections, the book I am studying from says that there are three diagrams who contribute to the amplitude, and they are:

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But other sources include also a fourth diagram, a tadpole-like one:

enter image description here

This seems to me a well-constructed diagram, because in the QCD Lagrangian there is the four-gluon vertex. Since the fourth diagram is not even mentioned in my book, I tried to prove that it gives no contribution to the gluon self-energy but I couldn't. Is it true? If yes (and I think it must be, otherwise the result presented in the book of the one-loop field-strength renormalization constant $Z_3$ is dead wrong), is there an easy way to prove it?

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    $\begingroup$ You can prove it directly by writing down the integral and seeing that it vanishes due to being a scaleless integral. $\endgroup$
    – Triatticus
    Commented Sep 23, 2022 at 19:29
  • $\begingroup$ Suggestion: Consider to call it a self-loop diagram for clarity. $\endgroup$
    – Qmechanic ♦
    Commented Sep 23, 2022 at 20:11
  • $\begingroup$ What do you mean when you say scaleless? Doing the calculation, I arrived to a point where (if they are correct, and they looked so to me) the amplitude is proportional to $$ \delta^{ab} \eta_{\alpha \beta} \int \frac{d^4k}{(2\pi)^4} \frac{1}{k^2+i\varepsilon}$$ Are you suggesting I should take the integral to be zero because it doesn't contain the mass term in the denominator? $\endgroup$
    – ar em
    Commented Sep 23, 2022 at 20:11
  • $\begingroup$ See here Dimensional Regularization and Massless Integrals $\endgroup$
    – Triatticus
    Commented Sep 27, 2022 at 13:33

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