The use of renormalisation constants often puzzles me. A good example is the use of Z_2 in the equation (7.58) of Peskin Schroeder. Z_2 is defined in equation (7.26).
as $Z_2^{-1} = 1-\frac{d\Sigma}{dp}$. Later in equation (7.31) it is said:

$Z_2-1 = d\Sigma/dp$ although this term is supposed to be infinite. But $d\Sigma/dp$ is treated of being smaller than 1. Okay, in this example the Pauli-Villars renormalisation is used where a rather high $\Lambda$ is needed to make $d\Sigma/dp$ larger than 1. But what would be if $Z_2$ were computed with dimensional regularization ?

Shouldn't be at least : $d\Sigma/dp + (d\Sigma/dp)^2 + (d\Sigma/dp)^3 + \dots $

I know that $d\Sigma/dp$ is of order $\alpha$ and there should be counter term to make the sum of both small (to order \alpha). On the other I am almost sure that when the counter terms of the next order \alpha^2 are calculated that it already forgotten that there was also a term $(d\Sigma/dp)^2$ which also need a counter term and for $\alpha^3$ order again and so on.
Could somebody explain it to me ? Thank you.

This post imported from StackExchange Physics at 2014-06-10 21:34 (UCT), posted by SE-user Frederic Thomas