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?


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