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In subleading powers of soft-collinear effective theory (SCET), the Lagrangian contains couplings between soft quarks and hard-collinear quarks. Matrix elements of the hard-collinear parts of these couplings are radiative jet functions. In the position-space formulation of SCET, the Lagrangians are constructed from operators that appear to be gauge invariant. Nevertheless, we find violations of gauge invariance arise in the hard-collinear sector because gauge transformations can shift the momentum of a hard-collinear quark field from the hard-collinear sector to the soft sector, where the hard-collinear fields, by definition, have no support. The violations of gauge invariance are manifested in perturbation theorymore » in the hard-collinear sector through the absence of certain Feynman diagrams that would be present in full QCD. A consequence of the absence of these diagrams is that the radiative jet functions that follow directly from the position-space Lagrangians are not gauge invariant, and we demonstrate this through explicit calculations in lower-order perturbation theory. We obtain gauge-invariant Lagrangians by adding to existing position-space Lagrangians terms that are proportional to the soft-quark equation of motion. These gauge-invariant Lagrangians are valid for nonzero, as well as zero, quark masses. We also remark briefly on the gauge invariance of certain Lagrangians that have been constructed in the label-momentum formulation of SCET. Published by the American Physical Society 2024« less

We propose to use transverse momentum p_T distribution of J/ψ production at the future Electron Ion Collider (EIC) to explore the production mechanism of heavy quarkonia in high energy collisions. We apply QCD and QED collinear factorization to the production of a $$c\bar{c}$$ pair at high p_T, and non-relativistic QCD factorization to the hadronization of the pair to a J/ψ. We evaluate J/ψ p_T-distribution at both leading and next-to-leading order in strong coupling, and show that production rates for various color-spin channels of a $$c\bar{c}$$ pair in electron-hadron collisions are very different from that in hadron-hadron collisions, which provides amore » strong discriminative power to determine various transition rates for the pair to become a J/ψ. We predict that the J/ψ produced in electron-hadron collisions is likely unpolarized, and the production is an ideal probe for gluon distribution of colliding hadron (or nucleus). We find that the J/ψ production is dominated by the color-octet channel, providing an excellent probe to explore the gluon medium in large nuclei at the EIC.« less