Computability of GPDs near $x=\pm \xi$ in Lattice QCD

In lattice QCD computations of generalized parton distributions (GPDs), the large momentum expansion generally requires all hard scales, $2\mathrm{\mid }x\pm \xi \mathrm{\mid }{P}^z$ and $2\mathrm{\mid }1\pm x\mathrm{\mid }{P}^z$, to be much larger than ${\mathrm{\Lambda }}_{\mathrm{Q}\mathrm{C}\mathrm{D}}$. We show that this condition can be relaxed for $2\mathrm{\mid }x\pm \xi \mathrm{\mid }{P}^z$ at large $\xi$, making the important $x\sim \pm \xi$ regions accessible to lattice calculations and considerably expanding the region of computability. Revisiting previous lattice results with complete one-loop matching, we obtain the expected partonic threshold behavior—GPDs continuous at $x=\pm \xi$ but with discontinuous derivatives—which has not previously been observed on the lattice. We thus obtain, for the first time, important prediction for GPDs in the distribution-amplitude-like region, which smoothly connects the quark and antiquark PDF-like behaviors.