We will identify $x_1$, ${\partial }_1$, and ${\gamma }_1$ with the Euclidean time direction; $\gamma$ matrices satisfy $\{{\gamma }_{\mu },{\gamma }_{\nu }\}=2{\delta }_{\mu \nu }$.

Although the theory given in Eq. (3) is often referred to as the Gross-Neveu model, it is not asymptotically free and chiral symmetry breaking requires a critical value for the coupling $g$ as in the original NJL model in 4D, and unlike the original Gross-Neveu model at large-$N$ in 2D; we will continue referring to it as the 3D NJL model at large $N$.

In this theory minimal subtraction MS means no subtraction, as there are no logarithmic divergences and hence no $1/(d-3)$ poles.

Since the functional $X[\varphi ]$ is typically nonlocal, by connected Feynman diagrams we mean neither conventional Feynman diagrams, nor conventionally connected. For example, expanding $X[\varphi ]$ to second order in $\varphi$ might take the form $(\int d^{3}xF(x)\varphi (x){)}^2$ for some function $F$; this is treated as a fundamental 2-point vertex in our discussion.