Direct quarkonium production in DIS from a joint CGC and NRQCD framework

We compute the differential cross section for direct quarkonium production in high-energy electron-nucleus collisions at small $x$. Our computation is performed within the nonrelativistic QCD factorization formalism that separates the calculation into short distance coefficients and long distance matrix elements that depend on the color and spin of the state. We obtain the short distance coefficients of the production of the heavy quark pair within the framework of the Color Glass Condensate effective field theory, which resums coherent multiple interactions of the heavy quark pair with the nucleus to all orders. Our results are expressed as the convolution of perturbatively calculable perturbative functions with multi-point light-like Wilson line correlators. In the correlation limit, we establish the correspondence between our CGC formulation with calculations employing the transverse momentum dependent (TMD) framework. We extend this correspondence by resumming kinematic power corrections within the improved TMD framework, which interpolates between the TMD formalism and $k_\perp$ factorization formalism. We present a detailed numerical analysis, focusing on $J/\psi$ production in the kinematics accessible at the future Electron-Ion Collider, highlighting the importance of genuine higher-order saturation contributions when the electron collides with a large nucleus.