The notion of renormalization
is probably one of the most difficult to understand and bizarre properties of the QFT. As for the renormalisation of couplings it seems strange, from the first sight, that the coupling constants in the bare Lagrangian of interacting theory mostly have to be infinite in order to get a finite results (amplitudes and cross-sections) for physical properties.
Renormalised fields and the bare fields are related by multiplicative way for the renormalizable theories: $$ \phi_R = Z_{\phi}^{1/2} \phi_0 $$ The same is true for mass, interaction coupling e.t.c. However, when in comes to the renormalisation of composite operators, renormalisation of an operator involves not a simple multiplication by some factor, but mixing with operators of the same dimension and properties under Lorentz transformations.
I am reading Collins'book Renormalization https://www.cambridge.org/core/books/renormalization/6EA5EEEBB9A02190F7C805856244181B. And in the 6th chapter the renormalisation of an operator is introduced by considering the $\phi^2$ operator for $\phi^3$ theory in 6D.
In order to deduce the expression for the renormalized operator, he is considering the Green's function: $$ \langle 0 | T \phi(x) \phi(y) \phi^2(z) | 0 \rangle $$ Then he considers all one-loop graphs for the theory. Some divergences are eliminated by the counterterms for the mass. But to kill the remaining divergences, he adds new counterterms with the same divergences, as those emerging, when performing integrations in the loops. And the resulting operator is: $$ \frac{1}{2} [\phi^2] = Z_{a} \frac{1}{2} \phi^2 + \mu^{d/2 - 3} Z_b m^2 \phi + \mu^{d/2 - 3} Z_c \phi $$ And the question is - how to correctly interpret the mixing of operators? Speaking roughly, I had a pig, but it turned out to have fish's scale, wings and horns, and it is in fact not a pig, but a strange hybrid, consisting of pig, deer, salmon and eagle. If a want to compute correlation functions with the given composite operator, QFT says:
Correlation function with $\phi^2$ is undefined, maybe you meant $\frac{1}{2}Z_a \phi^2 + \ldots$
Or, in another words, I cannot obtain any sensible result with the $\phi^2$, but there is a combination of $\phi^2$ and other stuff - $m^2 \phi$, $\Box \phi$ - that gives a finite result.
i apologize if the analogies are silly, I wanted to try to give myself some easy explanation.