- #1
fliptomato
- 78
- 0
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.
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?
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?