In Kaku's book, the self-energy in a [tex]\phi^4[/tex] scalar theory is expanded in a Taylor series as:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\Sigma(p^2)=\Sigma (m^2)+\Sigma'(m^2)(p^2-m^2)+\tilde_{\Sigma}(p^2) [/tex]

where [tex]\tilde_{\Sigma}(p^2) [/tex] is finite and m is arbitrary (but finite).

The full propagator is then:

[tex]i\Delta(p)=\frac{i}{p^2-m_{0}^2-\Sigma (m^2)-\Sigma'(m^2)(p^2-m^2)-\tilde_{\Sigma}(p^2)+i\epsilon} [/tex]

where m_{0}is the bare mass that's in the original Lagrangian. If we define [tex]m_{0}^2+\Sigma(m^2)=m^2 [/tex], i.e., the infinite bare mass cancels a divergence in a self-energy term to give something finite, then:

[tex]i\Delta(p)=\frac{i}{(1-\Sigma'(m^2))(p^2-m^2)-\tilde_{\Sigma}(p^2)+i\epsilon} [/tex]

Here's what I don't understand. Kaku now factors out a [tex] Z_\phi=\frac{1}{1-\Sigma'(m^2)} [/tex] to get:

[tex] i\Delta(p)=\frac{iZ_\phi}{(p^2-m^2)-\Sigma_{1}(p^2)+i\epsilon}[/tex]

where [tex]\Sigma_{1}(p^2) =Z_\phi\tilde_{\Sigma}(p^2)[/tex]

The [tex]Z_\phi[/tex] in the numerator of the propagator can be absorbed by bare constants, but I'm not sure how the [tex]Z_\phi[/tex] in the denominator (through [tex]\Sigma_1(p^2) [/tex]) can be gotten rid of.

Kaku defines the renormalized propagator [tex]\tilde{\Delta}(p)[/tex] as:

[tex]\Delta(p)=Z_\phi \tilde{\Delta}(p) [/tex]

which gets rid of [tex]Z_\phi[/tex] in the numerator, but not the denominator.

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# Renormalizing the propagator

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