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