$J/\psi$ Production Within Jets at the EIC

We present theoretical predictions for the transverse-momentum distribution of $J/\psi$ produced within jets at the upcoming Electron-Ion Collider (EIC). Utilizing the semi-inclusive fragmenting jet function (FJF) framework, our calculation achieves next-to-leading order (NLO) accuracy in the strong coupling and leading-logarithmic (LL) accuracy by resumming both collinear and threshold logarithms. In contrast to the gluon-dominated regime of the LHC, EIC photoproduction is characterized by an enhanced quark-initiated component, offering a complementary probe into the charmonium production mechanism governed by nonperturbative long-distance matrix elements (LDMEs). We examine the impact of representative LDME sets, demonstrating the EIC's distinct discriminating power for the mechanisms. We find that quark contributions are particularly significant in the small momentum fraction region. This region is also shown to be sensitive to both the jet radius $R$ and the experimental muon identification criteria for the $J/\psi \to \mu^+\mu^-$ decay channel. These findings establish quarkonium-in-jet observables at the EIC as a vital, independent probe for constraining the production mechanisms and advancing our understanding of heavy quarkonium formation.