Color-singlet relativistic correction to inclusive $J/\psi$ production associated with light hadrons at $B$ factories

We study the first-order relativistic correction to the associated production of $J/\psi$ with light hadrons at $B$ factory experiments at $\sqrt{s}=10.58$ GeV, in the context of NRQCD factorization. We employ a strategy for NRQCD expansion that slightly deviates from the orthodox doctrine, in that the matching coefficients are not truly of ``short-distance" nature, but explicitly depend upon physical kinematic variables rather than partonic ones. Our matching method, with validity guaranteed by the Gremm-Kapustin relation, is particularly suited for the inclusive quarkonium production and decay processes with involved kinematics, exemplified by the process $e^+e^-\to J/\psi+gg$ considered in this work. Despite some intrinsic ambiguity affiliated with the order-$v^2$ NRQCD matrix element, if we choose its value as what has been extracted from a recent Cornell-potential-model-based analysis, including the relative order-$v^2$ effect is found to increase the lowest-order prediction for the integrated $J/\psi$ cross section by about 30\%, and exert a modest impact on $J/\psi$ energy, angular and polarization distributions except near the very upper end of the $J/\psi$ energy. The order-$v^2$ contribution to the energy spectrum becomes logarithmically divergent at the maximum of $J/\psi$ energy. A consistent analysis may require that these large end-point logarithms be resummed to all orders in $\alpha_s$.