[Physics] Why does the counterterm’s propagator have inverse units of the propagator? $\phi^4$-theory


According to Peskin & Schroeder (page 325), the Feynman rule for the counterterm



$$ \frac12 \delta_Z(\partial_\mu\phi_r)^2-\frac12\delta_m \phi_r^2$$

being $\phi_r$ the renormalized field, is given by


which resembles rather the (multiplicative) inverse of the propagator for the original Lagrangian (whith physical quantities). Why?

Best Answer

Thee diverging terms for the propagator come from the renormalization of the self-energy $\Sigma$, defined by $G^{-1}=G_0^{-1}-\Sigma$, where $G_0$ is the propagator defined by the Lagrangian (i.e. bare propagator + counterterms) :

$G^{-1}_0=(1+\delta Z)p^2+(m^2_0+\delta m^2)$.

One chooses the counterterms to cancel the divergences coming from $\Sigma$ order by order. If the theory is perturbatively renormalizable, only these two counterterms are sufficient at every order in perturbation theory.

Related Question