Aren't $\phi^4$ composite operators? I have this trouble with terminology. I wonder why authors introduce the concept of composite operators after they've already talked about eg phi four theory, it phi cubed. Aren't these operators already composite ones? What's the difference? In particular when considering renormalization, say QED, the procedure goes like introducing factors of $Z^{1/2}$ for each field and so on. So why is the renormalization procedure in QED+Yukawa so different that one needs to consider how composite operators get renormalized? What's the big difference, isn't say $m\bar{\psi}\psi$ of QED already composite?
 A: Consider a correlator like
$$
\frac{1}{Z}\int \phi(x)^4 \phi(y_1)\ldots\phi(y_n)\ e^{-\int \phi(z)^4 dz}\ d\mu(\phi)
$$
where $d\mu$ is the perturbed free field measure. Both $\phi^4$'s are composite fields but very different ones. The $\phi(z)^4$ in the exponential is a composite field for the free field theory $d\mu$ whereas the $\phi(x)^4$ outside is a composite field for the full interacting theory. In fact, and this is Nelson's proof, it is possible to rigorously construct this interacting theory in 2d (in finite volume) by defining $\phi(z)^4$ as a $d\mu$ composite field (this is just the trivial Wick power renormalization) then sticking it in the exponential and integrating $d\mu(\phi)$.
Basic renormalization theory tells you how to get correlators
$$
\frac{1}{Z}\int \phi(x_1) \phi(x_2) \phi(x_3) \phi(x_4) \phi(y_1)\ldots\phi(y_n)\ e^{-\int \phi(z)^4 dz}\ d\mu(\phi)
$$
AFTER taking the limit $\Lambda\rightarrow\infty$ of removing the UV cut off, for non-coinciding points $x_1,\ldots,y_n$.
To get to the above correlator with $\phi(x)^4$
you still have to follow this by the analysis of the very singular limit $x_1,x_2,x_3,x_4\rightarrow x$. This needs more advanced renormalization theory, typically in the following chapters of QFT textbooks about composite fields and the OPE.
The process I described is in fact the construction of the composite field $\phi^4$
in the interacting theory using the OPE. An alternative construction (in fact the more standard one) is to take the limit $x_1,x_2,x_3,x_4\rightarrow x$ first and then take $\Lambda\rightarrow\infty$. Both procedures should give the same result.
PS: I gave a talk on this just last week at MSU, for a math audience. See
https://www1.math.msu.edu/seminars/gauge_theory/
