Connecting Euclidean to light-cone correlations: From flavor nonsinglet in forward kinematics to flavor singlet in non-forward kinematics

We present a unified framework for the perturbative factorization connecting Euclidean correlations to light-cone correlations. Starting from nonlocal quark and gluon bilinear correlators, we derive the relevant hard-matching kernel up to the next-to-leading-order, both for the flavor singlet and non-singlet combinations, in non-forward and forward kinematics, and in coordinate and momentum space. The results for the generalized distribution functions (GPDs), parton distribution functions (PDFs), and distribution amplitudes (DAs) are obtained by choosing appropriate kinematics. The renormalization and matching are done in a state-of-the-art scheme. We also clarify some issues raised on the perturbative matching of GPDs in the literature. Our results provide a complete manual for extracting all leading-twist GPDs, PDFs as well as DAs from lattice simulations of Euclidean correlations in a state-of-the-art strategy, either in coordinate or in momentum space factorization approach.