# Field Strength Renormalisation in Peskin and Schroeder

In chapter 7 of Peskin and Schroeder they define the field strength renormalisation $Z$ for a quantum field to be the residue of the Fourier transform of the correlation function

$$\langle \Omega | \phi(x)\phi(0) | \Omega\rangle$$

They go on to refer to $Z$ variously as $Z_1$, $Z_2$ and $Z_3$ at different points in the text. Is the subscript meant to denote the order to which we've computed $Z$? In other words do we have

$$Z = Z_1 + O(e^2) = Z_2 + O(e^3) = Z_3 + O(e^4)$$

and so on? This would seem to make no sense because then naturally $Z_1 = Z_2$ due to the Feynman diagram stucture.

Alternatively are they just meant to be $Z$s for different systems (Lagrangians)? Or are they just genuinely different quantities depending on context?

I'm getting confused because P&S generally use subscripts to denote the order to which a quantity is valid, but it doesn't seem to be the case here!

If anyone has the book and wants page references, see p215, 221 and 243.

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No. $Z$, $Z_2$ and $Z_3$ is the field strength renormalization for $A_\mu$ and $\psi$ respectively. $Z_1 = \frac{e_0}{e_R}Z_2 \sqrt{Z_3}$ where $e_R$ and $e_0$ are the renormalized and the bare couplings respectively. –  Prahar Aug 1 '13 at 22:40