Confusion about the Z factor(Renormalization factor)

[phylz]

In Peskin's textbook chapter 7 Radiative Corrections: Some formal developments (page 229), he said the Z factors are irrelevant for calculations at the leading order of perturbation theory, but are important in the calculation or higher-order corrections.

My question is how can the Z factor be irrelevant for calculations at the leading order of perturbation theory. Thanks.

 

[phylz]

In Peskin's textbook chapter 7 Radiative Corrections: Some formal developments (page 229), he said the Z factors are irrelevant for calculations at the leading order of perturbation theory, but are important in the calculation or higher-order corrections.

My question is how can the Z factor be irrelevant for calculations at the leading order of perturbation theory. Thanks.

Z is the field-strength renormalization factor

 

[DarMM]

At the lowest order in perturbation theory it is just $1$.

 

[phylz]

At the lowest order in perturbation theory it is just $1$.

@DarMM thank you. Now I got it, for electron, $Z_2^(-1)=1-\frac{d\Sigma}{d\slashed{p}}|_{\slashed{p}=m}$, at the leading-order contribution, electron self-energy is 0, so Z=1.

 

[DarMM]

Yes exactly. Note that the singularities in $Z$ and hence the need to renormalise it arise from the fact that $Z$ is not analytic in the coupling constant, so Taylor expanding it causes divergences.