# Amputated Green's function in the LSZ formula

From Schwartz's QFT textbook, under Ch.18, Mass renormalization, Schwartz introduces a new LSZ formula with renormalized Green's function. He states that the new LSZ formula for QED, with pole mass $m_p$, is $$\langle f|S|i\rangle = (\not\!{p_f} - m_p)\dots (\not\!{p_i} - m_p) \langle\psi^R\dots\psi^R\rangle_{\text{amputated}}$$ where amputated diagrams signify external lines chopped off until they begin interacting with the other fields.

But don't we get amputated diagram after we apply the $(\not\!{p_j}-m_p)$ onto the Green's function to project it to $S$-matrix? I don't understand why external lines are already 'amputated' even before applying $(\not\!{p_j}-m_p)$.

Also, another side question, is pole mass defined to be physical mass by definition? I don't understand what forces pole mass to be the actual observable physical mass.

• Yeah, that's why I wrote "(sorry I don't know how to type Dirac slash.)"... Hopefully someone can edit it for me. Jun 11 '15 at 6:36

The overall factors $(\not{p} - m)$ are indeed responsible for amputating the external legs. I don't have a copy of Schwartz on hand so I can't comment on what he might have meant, but the correlation functions do have propagators on the external legs before putting them on shell by applying LSZ.
The mass that goes into the LSZ formula is the physical, observable mass (so long as you normalize everything properly)--the mass of the asymptotic state. You can see this if you go through the LSZ derivation. The reason that the pole of the propagator corresponds to the physical mass is precisely because when the LSZ formula puts external particles on shell, it "picks out" the poles in the propagators (since you have $(\not{p} - m)\times 1 / (\not{p} - m)$), at the physical mass.