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I am reading "Quantum Field theory and the Standard Model" by Schwartz. It derives LSZ formula in chapter 6,

\begin{equation} \langle f|S|i\rangle =\left[i\int d^4x_1e^{-ip_1x_1}(\Box + m^2)\right]...\left[i\int d^4x_ne^{-ip_nx_n}(\Box + m^2)\right]\times\langle\Omega|T\{\phi(x_1)\phi(x_2)\phi(x_3)...\phi(x_n)\}|\Omega\rangle \end{equation}

After that the book gives examples of tree-level calculations for scattering amplitude of some simple QED process. So far so good.

But then in later chapters the book talks about renormalization where I am told that the $m$ in Lagrangian isn't actually the pole mass, but is in fact infinite so it cancels some infinities from loop diagrams. I sort of understand how $m$ in Lagrangian isn't an observable so it doesn't have to be the pole mass, but then how do we know that the value of $m$ in LSZ formula must be pole mass? I re-read the derivation of LSZ formula in Schwartz and the only justification seems to be that at $t=\pm\infty$, the field $\phi$ satisfies the non-interacting Klein-Gordon equation, but doesn't the mass $m$ in Klein-Gordon equation comes from Lagrangian, which should be infinite?

I appreciate your help. Thanks a lot in advance.

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Because in LSZ you have $(\partial^2 + m^2)$ and the propagator in momentum space is $\frac{i}{-k^2 + m^2}$ so you can see that when a plane wave acts on the differential operator you get something proportional to 1/propagator, so that will cancel out the propagators to get a finite result, which is only the case if the $m$ that shows in the differential operator is the pole $m$.

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