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The production cross-sections of $J/\psi$ mesons in proton-proton collisionsat a centre-of-mass energy of $\sqrt{s}=5$ TeV are measured using a data samplecorresponding to an integrated luminosity of $9.13\pm0.18~\text{pb}^{-1}$,collected by the LHCb experiment. The cross-sections are measureddifferentially as a function of transverse momentum, $p_{\text{T}}$, andrapidity, $y$, and separately for $J/\psi$ mesons produced promptly and frombeauty hadron decays (nonprompt). With the assumption of unpolarised $J/\psi$mesons, the production cross-sections integrated over the kinematic range$0<p_{\text{T}}<20~\text{GeV}/c$ and $2.0<y<4.5$ are$8.154\pm0.010\pm0.283~\mu\text{b}$ for prompt $J/\psi$ mesons and$0.820\pm0.003\pm0.034~\mu\text{b}$ for nonprompt $J/\psi$ mesons, where thefirst uncertainties are statistical and the second systematic. Thesecross-sections are compared with those at $\sqrt{s}=8$ TeV and $13$ TeV, andare used to update the measurement of the nuclear modification factor inproton-lead collisions for $J/\psi$ mesons at a centre-of-mass energy pernucleon pair of $\sqrt{s_{\text{NN}}}=5$ TeV. The results are compared withtheoretical predictions.

Measurement of $J/\psi$ production cross-sections in $pp$ collisions at $\sqrt{s}=5$ TeV