Constraints on Nonrelativistic-QCD Long-Distance Matrix Elements from $J/\psi$ Plus $W$/$Z$ Production at the LHC

We study the associated production of prompt $J/\psi$ mesons and $W$ or $Z$ bosons within the factorization approach of nonrelativistic QCD (NRQCD) at next-to-leading order in $\alpha_s$, via intermediate color singlet ${^3}S_1^{[1]}$ and ${^3}P_J^{[1]}$ and color octet ${^1S}_0^{[8]}$, ${^3S}_1^{[8]}$ and ${^3P}_J^{[8]}$ states. Requiring for our predictions to be compatible with recent ATLAS measurements yields stringent new constraints on charmonium long-distance matrix elements (LDMEs) being nonperturbative, process-independent input parameters. Considering four popular LDME sets fitted to data of single $J/\psi$ inclusive production, we find that one is marginally compatible with the data, with central predictions typically falling short by a factor of three, one is unfavored, the factor of shortfall being about one order of magnitude, and two violate cross section positivity for direct $J/\psi+W/Z$ production. The large rate of prompt $J/\psi$ plus $W$ production observed by ATLAS provides strong evidence for the color octet mechanism inherent to NRQCD factorization, the leading color singlet contribution entering only at $\mathcal{O}(G_F\alpha_s^4)$, beyond the order considered here.