I have been working on the full renormalization of scalar QED with self-interactions, following the steps of Schwartz’s treatment on spinor QED (Chap 19). I have 3 main questions regarding this:
Need I show that the 3-point function is UV finite (which coincidentally shows $Z_1=Z_2$), or can I use the fact that $Z_1=Z_2$ is easy to prove by gauge invariance to avoid that calculation? I think, as it is proven in section 19.5, using gauge invariance, $Z_1=Z_2$ is only a necessary condition for vertex renormalization. I don’t fully understand the longer proof he gives in 19.5.1, so I can’t say if it solves my issue, but my understanding is that I still have to show that $Z_1$ indeed cancels the UV divergence.
When calculating the counterterms for 4-point functions, do I need to compute separately the $s$, $t$, and $u$ channels of a process, or do they surely all cancel with the same counterterm? Why? My understanding is I have to compute them all.
In the 4 scalar vertex, to conserve particle number, we need two inwards arrows, and two outwards, but does the order of arrows matter? Going around, in-out-in-out is a different vertex from in-in-out-out. Does it matter? Both appear in literature. I think it does, and I think the former is correct given the Lagrangian.
