Tadpole graph & normal ordering

[QuantumDevil]

It is said that so called tadpole graph gets eliminated from perturbation expansion if the normal-ordered interaction is adopted. How can it be proved? Can anybody provide some links or any other references about this problems?

$$L_{int}=-e:\bar{\psi}\gamma_{\mu}\psi A^{\mu}:$$

 

[QuantumDevil]

Here is attachement with this graph

 

[QuantumDevil]

...and this also should remove the mass shift I suppose.

 

[blechman]

See Peskin and Schroder, Chapter 4.

You can see right away that (at least in QED) tadpoles for the photon must vanish, since QED has a charge-conjugation symmetry in which the photon field is odd, so the one-point function for the photon must vanish. You can also see that the diagram you included vanishes since the loop integral explicitly vanishes:

$$\int d^4k\frac{{\rm Tr}[\gamma^\mu(k\!\!\!\slash+m)]}{k^2-m^2}= \int d^4k\frac{4k^\mu+0}{k^2-m^2}=0$$

These two statements are, of course, related.

 

[loopist]

How do you know that it is zero by direct computation?