Color-singlet relativistic correction to inclusiveJ/ψproduction associated with light hadrons atBfactories

Abstract

We study the first-order relativistic correction to the associated production of $J/\ensuremath{\psi}$ with light hadrons at $B$ factory experiments at $\sqrt{s}=10.58\text{ }\text{ }\mathrm{GeV}$, in the context of nonrelativistic QCD (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 a ``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 ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}J/\ensuremath{\psi}+gg$ 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 $J/\ensuremath{\psi}$ cross section by about 30%, and exert a modest impact on $J/\ensuremath{\psi}$ energy, angular and polarization distributions except near the very upper end of the $J/\ensuremath{\psi}$ energy. The order-${v}^{2}$ contribution to the energy spectrum becomes logarithmically divergent at the maximum of $J/\ensuremath{\psi}$ energy. A consistent analysis may require that these large end-point logarithms be resummed to all orders in ${\ensuremath{\alpha}}_{s}$.