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The ATLAS collaboration recently reported on the first observation ofassociated-production of a $Z^0$ boson with a $J/\psi$. We recently claimedthat the corresponding yield of the {\it prompt} $J/\psi$ was dominated bydouble parton scatterings in the ATLAS acceptance with a somewhat small valueof $\sigma_{\rm eff}$. We also found out that single parton scatterings wereonly dominant at large transverse momenta. We present here the firstphenomenological analysis of another part of the ATLAS data sample, namely of a$Z^0$ boson plus a {\it non-prompt} $J/\psi$. Our study is performed atnext-to-leading order in $\alpha_s$ and includes parton-shower effects via the{\sc\small MadGraph5_aMC@NLO} framework. We find out that the data, unlike thecase of prompt $J/\psi+Z^0$, do not hint at significant DPS contributions.Owing to the current experimental and theoretical uncertainties, there is stilla room for these but with a lower limit of $\sigma_{\rm eff}$ close to 5 mb. Westress the importance of QCD corrections to account for the ATLAS data.

Phenomenological analysis of associated production of $Z^0+b$ in the $b \rightarrow J/\psi X$ decay channel at the LHC