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The associated production of prompt $J/\psi$ and $\mathit{\mathit{\Upsilon}}$mesons in $pp$ collisions at a centre-of-mass energy of$\sqrt{s}=13\,\mathrm{TeV}$ is studied using LHCb data, corresponding to anintegrated luminosity of $4\,\mathrm{fb}^{-1}$. The measurement is performedfor $J/\psi$ ($\mathit{\Upsilon}$) mesons with a transverse momentum$p_{\mathrm{T}}<10\,(30)\,\mathrm{GeV}/c$ in the rapidity range $2.0<y<4.5$. Inthis kinematic range, the cross-section of the associated production of prompt$J/\psi$ and $\mathit{\Upsilon}(1S)$ mesons is measured to be $133 \pm 22 \pm 7\pm 3 \, \mathrm{pb}$, with a significance of $7.9\,\sigma$, and that of prompt$J/\psi$ and $\mathit{\Upsilon}(2S)$ mesons to be $76\pm 21 \pm 4 \pm 7 \,\mathrm{pb}$, with a significance of $4.9\,\sigma$. The first uncertainty isstatistical, the second systematic, and the third due to uncertainties on theused branching fractions. This is the first observation of the associatedproduction of $J/\psi$ and $\mathit{\Upsilon}(1S)$ in proton-proton collisions.Differential cross-sections are measured as functions of variables that aresensitive to kinematic correlations between the $J/\psi$ and$\mathit{\Upsilon}(1S)$ mesons. The effective cross-sections of the associatedproduction of prompt $J/\psi$ and $\mathit{\Upsilon}$ mesons are obtained andfound to be compatible with measurements using other particle productions.

Associated production of prompt $J/\psi$ and $\mathit{\Upsilon}$ mesons in $pp$ collisions at $\sqrt{s}=13\,\mathrm{TeV}$