Peskin and Schroeder Exercise 10.2 - Yukawa Theory Renormalization

Question

I am having some trouble with exercise 10.2 in Peskin and Schroeder, on the renormalization of Yukawa theory. Part a) of the exercise says

show that the theory contains a superficially divergent 4$\phi$ amplitude. This means that the theory cannot be renormalized unless one includes a scalar self-interaction $\frac{\lambda}{4!} \phi^4$ and a counterterm of the same form.

It makes sense to me that this would be one way to absorb the infinities, but why is adding that extra term the only way to renormalize the theory?

In other words, why exactly is the 4$\phi$ amplitude special, and could not be renormalized (absorbed/taken care of) if we didn't add that extra term, used renormalized perturbation theory to derive Feynman rules for the counterterm vertices where we would have an analogous vertex with a counterterm coupling which we could also use to obtain a counterterm version of the 4$\phi$ coupling?

I didn't write the whole thing out, so I guess we just wouldn't have enough terms to "match" all the infinities, but the way it's stated in the task and presented in the sample solutions I found online leads me to believe that there may be a quicker way to just "see" that we need this extra term in the Lagrangian before we even start with the perturbation theory procedure.

Answer 1

The rule of thumb is that if you have a 1-loop N particle scattering process that diverges, then their must be a corresponding tree level term in the lagrangian that generates the same N particle scattering process, so that the divergence can be absorbed by renormalizing the couplings of the tree level term.

You can see a consequence of this in QED, which is the physical theory of electrons and photons. If you consider the scattering of 4 photons, this can occur only at one loop. Yet there is no tree level term in the QED lagrangian that generates a 4 photon scattering amplitude. This means that the loop diagram that generates this must NOT diverge. I hope this answers your question.