The production of a $Z$ boson in association with a $J/\psi$ meson in proton-proton collisions probes the production mechanisms of quarkonium and heavy flavour in association with vector bosons, and allows studies of multiple parton scattering. Using $20.3\, \mathrm{fb^{-1}}$ of data collected with the ATLAS experiment at the LHC in $pp$ collisions at $\sqrt{s}=8\, \mathrm{TeV}$, the first measurement of associated $Z + J/\psi$ production is presented for both prompt and non-prompt $J/\psi$ production, with both signatures having a significance in excess of $5\sigma$. The inclusive production cross-sections for $Z$ boson production (analysed in $\mu^+\mu^-$ or $e^+e^-$ decay modes) in association with prompt and non-prompt $J/\psi(\to\mu^+\mu^-)$ are measured relative to the inclusive production rate of $Z$ bosons in the same fiducial volume to be $(36.8\pm 6.7\pm 2.5) \times 10^{-7}$ and $(65.8\pm 9.2\pm 4.2) \times 10^{-7}$ respectively. Normalised differential production cross-section ratios are also determined as a function of the $J/\psi$ transverse momentum. The fraction of signal events arising from single and double parton scattering is estimated, and a lower limit of $5.3\ (3.7)\, \mathrm{mb}$ at $68\ (95)$% confidence level is placed on the effective cross-section regulating double parton interactions.
The fiducial, inclusive (SPS+DPS) and DPS-subtracted differential cross-section ratio $R_{Z+J/\psi}$ for prompt $J/\psi$.
The fiducial, inclusive (SPS+DPS) and DPS-subtracted differential cross-section ratio $R_{Z+J/\psi}$ for non-prompt $J/\psi$.
The fiducial, inclusive (SPS+DPS) and DPS-subtracted differential cross-section ratio $\mathrm{d}R_{Z+J/\psi}/\mathrm{d}y$ as a function of $y_{J/\psi}$ for prompt $J/\psi$.
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