I have a very basic question about Renormalization in Quantum Field Theory. Consider the following passage (about $\phi^4$ theory) from Zee's Quantum Field Theory in a Nutshell (from Chapter III.3):
Shouldn't we start out with a zeroth order theory already written in terms of the physical mass $m_{P}$ and physical coupling $\lambda_{P}$ that experimentalists actually measure, and perturb around that theory? Yes, indeed, and this way of calculating is known as renormalized or dressed perturbation theory, or as I prefer to call it, physical perturbation theory. We write $$ \mathcal{L} = \tfrac{1}{2} (\partial \phi)^2 - \tfrac{1}{2}m_{P}^2 \phi^2 - \frac{\lambda_{P}}{4!}\phi^4 + A (\partial \phi)^2 + B \phi^2 + C \phi^4 \ . $$
The factors $A$, $B$, $C$ are calculated iteratively in perturbation theory and designed at each order to cancel out the divergences that occur in the loop computations.
My question is, if we perturb about the physical theory, what justifies the use of perturbation theory? I thought that perturbation theory is only justified when the corrections to the free theory are "small" $\to$ here this doesn't seem to be the case to me since the ``couplings'' $A$, $B$, $C$ don't seem to be small, as I understand it they are divergent quantities which are functions of the cutoff/regulator used ($A$ and $C$ are dimensionless, so I think they are actually just large numbers - while $B$ has mass dimension 1 so I actually don't know what it means for $B$ to be small here).
