Hello--I'm looking at Peskin p.324-323 where he describes the renormalization of [tex]\phi^4[/tex] theory. I'm a little confused about the Feynman rules that one gets out of the lagrangian with counter terms.(adsbygoogle = window.adsbygoogle || []).push({});

My question in a nutshell: The propagator is given by [tex]\frac{i}{p^2 - m^2}[/tex], why is it that the counter term looks like the inverse of this, namely [tex]i(p^2\delta_Z - \delta_m)[/tex], when they come from terms in the lagrangian that have identical form?

That is to say, the lagrangian contains the terms:

[tex]\frac{1}{2}(\partial_\mu \phi_r)^2 - \frac{1}{2}m^2\phi_r^2 + \frac{1}{2}\delta_Z(\partial_\mu \phi_r)^2 -\frac{1}{2}\delta_m\phi_r^2[/tex]

The first two yield the propagator with junk in the denominator, while the last two yield a counterterm with junk in the numerator.

Why is this?

I can "read off" the rules for the counter terms, and it makes sense that it puts stuff in the numerator. Similarly, I know that the propagator is given by the Green's function of the free theory, which is why it yields something in the denominator. But am I thinking about it too much if I think it's really strange?

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# Propagator counterterm in phi^4

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