Renormalization in $\lambda\phi^4$-theory: Why renormalize at one-loop instead of renormalizing at order of the coupling constant $\lambda$?

Question

I am reading about one-loop renormalization in the $\lambda\phi^4$-theory. Instead of doing renormalization at order $\lambda$, why are we interested in renormalization at one-loop which contains both order $\lambda$ and order $\lambda^2$ diagrams? More generally, instead of renormalizing at any fixed order of perturbation theory, why do we renormalize at one-loop or two-loop or three-loop, etc?

Answer 1

OP is right that in perturbative QFT we ultimately want to calculate in order of the coupling constant $\lambda$, e.g. to determine the beta function. However, the following facts should be kept in mind:

1. Since the underlying relevant Feynman diagrams (that needs to be calculated) are 1PI, the order of coupling constants (which is the order $V$ of vertices) is typically closely tied to the order $L$ of loops.

2. In practice the complexity of a calculation grows rapidly with the number $L$ of loop-momentum integrations, so it is custom to label in terms of $L$ rather than $V$. (Moreover, the loop expansion makes independently sense as an expansion in Planck's constant $\hbar$, cf. e.g. my Phys.SE answer here.)