# Meson operator treated as an elementary field in effective action?

I have seen in many papers, with this as the earliest example I can find, that the meson chiral order operator $$M_j^{\,\,\,i}(x)=\overline{\psi}^i(1+\gamma_5)\psi_j(x)$$ is treated as an elementary operator (as opposed to a composite one) when sourcing it to construct an effective potential. The effective potential $$V(M)$$ is then a general function of $$\text{tr}(MM^{\dagger})^n$$ for spacetime independent $$M$$.

Shouldn't this more rigorously be treated as a composite operator $$M_j^{\,\,\,i}(x,y)$$, and construct the $$2$$PI effective action according to this paper? It is my understanding that the effective potential for a composite operator is in general much more complicated than that of an elementary operator, since translation invariance only simplifies the composite operator to depend on $$x-y$$. This then gives room for complicated spacetime dependent kernels in between each $$M_j^{\,\,\,i}(x-y)$$.

Under what circumstances are we allowed to treat a composite operator as an elementary field?