Unraveling Gluon TMDs in $J/\psi$ and Pion production at the EIC

We investigate the azimuthal asymmetries such as $\cos2{\phi_T}$ and Sivers symmetry for $J/\psi$ and $\pi^\pm$ production in electron-proton scattering, focusing on scenarios where the $J/\psi$ and the pion are produced in an almost back-to-back configuration. The electron is unpolarized, while the proton can be unpolarized or transversely polarized. For the $J/\psi$ formation, we use non-relativistic QCD (NRQCD), while $\pi^\pm$ is formed due to parton fragmentation. In this kinematics, we utilize the transverse momentum-dependent factorization framework to calculate the cross sections and asymmetries. We consider both quark and gluon-initiated processes and show that the gluon contribution dominates. We provide numerical estimates of the upper bounds on the azimuthal asymmetries, as well as employ a Gaussian parametrization for the gluon transverse momentum distributions (TMDs), within the kinematical region accessible by the upcoming Electron-Ion Collider (EIC).