LSZ formula and Field Strength renormalization factor

[Luca_Mantani]

Hi all,
I have doubts on the role of the Field Strength renormalization $Z_\psi=1+\delta_\psi$ when computing amplitudes. I never did this, maybe because it was not needed before, but i noticed that in the solution of a specific problem, to obtain the correct result, you need to multiply the sum of the amplitudes by $\sqrt{Z_\psi}$ for every external leg. So it seems that the correct formula, for a 2 to 2 scattering for example, is:
$$Z_\psi^2(A_1+A_2+ ... + A_n)$$
where $A_i$ are all the possible connected and amputed diagrams and they also include all the countertems diagrams, is that right?

So to be clear, if i have to compute scattering amplitude in QED for $e^- \mu^- \to e^- \mu^-$ i sum the tree level diagram with the fermions exchanging a photon, the diagrams with the loop in the vertex, the diagram with the loop in the photon, the diagram with the counterterm for the vertex and the diagram with the counterterm for the photon self-energy. Then i multiply everything for $Z_\psi^2$.

Is that right?

 

[vanhees71]

Yes, that's right. Have a look at Bjorken, Drell Vol. 2, where this is discussed with some more care than in many modern textbooks.