Graduate S-matrix expansion and Wick's theorem

[Gaussian97]

TL;DR

What is the correct way to write the S-matrix?
$$S = T\left\{e^{-i\int \mathscr{H}_I d^4 x}\right\}$$
or
$$S = T\left\{e^{-i\int :\mathscr{H}_I: d^4 x}\right\}$$
?
Where :: refers to the Normal-ordering.

My question arises when we expand the S-matrix using Wick's theorem, there we need to compute time-ordered products, but is not the same to compute
$$T\{\bar{\psi}(x_1)\gamma^\mu \psi(x_1) A_\mu(x_1)\}$$ or
$$T\{:\bar{\psi}(x_1)\gamma^\mu \psi(x_1) A_\mu(x_1):\}$$
While the second one is simply
$$:\bar{\psi}(x_1)\gamma^\mu \psi(x_1) A_\mu(x_1):$$
the first one is
$$T\{\bar{\psi}(x_1)\gamma^\mu \psi(x_1) A_\mu(x_1)\} = :\bar{\psi}(x_1)\gamma^\mu \psi(x_1) A_\mu(x_1): + i \text{Tr}\{{S_F(0)\gamma^\mu\}}A_\mu(x_1)$$
So there's a difference between both, and similar diferences appear in higher order calculations, now I know that this extra terms give tadpole diagrams and that this vanish in QED. But in other theories we need to introduce this terms?

Thanks

 

[vanhees71]

Tadpole diagrams are constant self-energy insertions, which are renormlized away anyway. In gauge theories it's convenient to keep them in, because then in gauge-invariant renormalization schemes like dimensional regularization the Ward Takahashi identities are fulfilled at any order for the regularized Feynman diagrams. E.g., in scalar QED at the one-loop level the photon self-energy is only transverse when you take the tadpole diagram into account. Normal ordering cancels the tadpole diagram but this is obviously not a gauge-invariant procedure. Of course this doesn't really matter, you only have to keep in mind to choose your counter terms at any order PT such that the WTIs stay fulfilled for the renormalized proper vertex functions.