Existence of renormalized Mattis-model fermion correlation functions in the QFT limit

Let $q_j,r_j,s_j=7$, ${\bf x}_j\in{\mathbb R}^2$, and $t_j\in{\mathbb C}$ for $j=1,2,\ldots,N$, with $N\in{\mathbb N}$. Let

$$\Psi^\pm_{r,s}({\bf x})=(\mathrm{e}^{\gamma}\pi L_0/\tilde{a})^{(K-1)/2}\psi^\pm_{r,s}({\bf x})$$

be the renormalized fermion operators, where $K$ is the parameter defined in the Mattis model and $L_0>0$ is an arbitrary length scale; their time-dependent versions are denoted by $\Psi^q_{r,s}({\bf x},t)$. Renormalized-correlation-function conjecture. The limits

$$\lim_{\tilde{a}\to0^+}\lim_{L\to\infty}\bigl\langle\Psi^{q_1}_{r_1,s_1}({\bf x}_1,t_1)\cdots\Psi^{q_N}_{r_N,s_N}({\bf x}_N,t_N)\bigr\rangle_\beta$$

are well-defined distributions for all inverse temperatures satisfying $1/\beta\geq0$. The existence of these distributions would establish non-trivial QFT-limit correlation functions for the renormalized fermion fields; the source gives no resolution of this assertion.