In page 319 in Peskin's An Introduction to Quantum Field Theory, renormalization of QED is discussed, and it's shown that there are only four divergent quantities involved. But this conclusion is based on the assumption that potentially divergent amplitude is an analytic function of the external momenta, for example for the electron self energy $i\Sigma(\displaystyle{\not p})=A_0+A_1\displaystyle{\not p}+A_2 \displaystyle{\not p}^2+\ldots$. Higher order coefficients in the Taylor series are more and more convergent by simple power counting as is argued in the book, hence we always get a finite number of divergent quantities in an amplitude. However, I can't see an obvious reason why the amplitude should be an analytic function of the external momenta. In my simple mind free from math rigor now, I think this would be true if the internal propagators can't go on-shell, but I can't see why this must be true either.
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