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In Section 3.7.2 of Tong's QFT notes the LSZ reduction formula is briefly discussed.

Essentially, this tells us that for S-matrix elements we can use the same momentum space Feynman rules as for correlation functions, except that:

  1. We should remove all external line propagators,

  2. We should place the corresponding momenta back on the mass-shell.

Since for correlation functions all disconnected 'bubble' diagrams cancel, we should also only consider connected Feynman diagrams.

However, there are two things missing from this discussion that I see in other texts:

  1. There is no field strength renormalisation factor $\sqrt{Z}$,

  2. There is nothing that says we need to consider amputated diagrams only.

I would like to know, roughly, how these two effects arise from the LSZ reduction formula presented in these notes. Or, is it the case that these are constraints imposed after the derivation of the formula, for some physical reason?

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  1. The factors of $\sqrt Z_i$ are typically implicit, or more precisely, they are reabsorbed into the fields $\phi_i$. To get these factors back, you just have to rescale $\phi_i\to \sqrt Z_i\phi_i$.

  2. In the LSZ formula you multiply the external lines by the factor $p^2+m^2$, and then take $p^2\to-m^2$ to put these lines on-shell. This automatically amputates all external lines, because any loop correction vanishes at $p^2=-m^2$ (due to the renormalisation condition $\Pi(-m^2)=0$).

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