I am wondering if it is always certain that there are enough counterterms to renormalize a renormalizable (e.g. non-negative mass dimension of coupling constant) theory. Through some methods such as power counting, one finds that there are a certain number of divergent parameters, which must be absorbed into free parameters of the Lagrangian, using counterterms. However, in the examples I have seen, there just so happens to be as many counterterms as divergent parameters to make this possible.
For example, in $\phi^4$ theory (neglecting vacuum diagrams), there are two divergent numbers in the two-point function, and one in the four-point function, which are able to be absorbed into the bare mass, the bare coupling constant, and the field strength renormalization. Similarly, there are four divergent numbers in QED which are absorbed into the electron mass and charge, and the electron and photon field strength renormalization.
Given that the arguments to determine how many numbers are divergent are not entirely trivial, why should we expect that there will be enough free parameters to absorb them into, even in a renormalizable theory?
Best Answer
This is probably easiest to see using the Wilsonian effective action. To ensure that a counterterm is available, we should include it in the action to begin with.
(If we are discussing renormalizability in the Dyson sense, then we only allow relevant and marginal action terms, i.e. action terms with coupling constants of non-negative mass dimension, cf. e.g. this Phys.SE post.)
For examples, see e.g. this and this Phys.SE posts.