Bjorken and threshold asymptotics of a space-like structure function in the 2D $U(N)$ Gross-Neveu model

In this work, we investigate a coordinate space structure function ${\cal E}(z^2m^2,\lambda)$ in the 2D $U(N)$ Gross-Neveu model to the next-to-leading order in the large-$N$ expansion. We analytically perform the twist expansion in the Bjorken limit through double Mellin representations. Hard and non-perturbative scaling functions are naturally generated in their Borel representations with detailed enumerations and explicit expressions provided to all powers. The renormalon cancellation at $t=n$ between the hard functions at powers $p$ and the non-perturbative functions at powers $p+n$ are explicitly verified, and the issue of ``scale-dependency'' of the perturbative and non-perturbative functions is explained naturally. Simple expressions for the leading power non-perturbative functions are also provided both in the coordinate space and the momentum-fraction space ($0<\alpha<1$) with ``zero-mode-type'' subtractions at $\alpha=0$ discussed in detail. In addition to the Bjorken limit, we also perform the threshold expansion of the structure function up to the next-to-next-to-leading threshold power exactly and investigate the resurgence relation between threshold and ``Regge'' asymptotics. We also prove that the twist expansion is absolutely convergent for any $0<z^2<\infty$ and any $ \lambda \in iR_{\ge 0}$.