The production cross-sections of $J/\psi$ mesons in proton-proton collisions at a centre-of-mass energy of $\sqrt{s}=5$ TeV are measured using a data sample corresponding to an integrated luminosity of $9.13\pm0.18~\text{pb}^{-1}$, collected by the LHCb experiment. The cross-sections are measured differentially as a function of transverse momentum, $p_{\text{T}}$, and rapidity, $y$, and separately for $J/\psi$ mesons produced promptly and from beauty 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 the first uncertainties are statistical and the second systematic. These cross-sections are compared with those at $\sqrt{s}=8$ TeV and $13$ TeV, and are used to update the measurement of the nuclear modification factor in proton-lead collisions for $J/\psi$ mesons at a centre-of-mass energy per nucleon pair of $\sqrt{s_{\text{NN}}}=5$ TeV. The results are compared with theoretical predictions.
The production of $B^{\pm}$ mesons is studied in $pp$ collisions at centre-of-mass energies of 7 and 13 TeV, using $B^{\pm}\rightarrow J/\psi K^{\pm}$ decays and data samples corresponding to 1.0 fb$^{-1}$ and 0.3 fb$^{-1}$, respectively. The production cross-sections summed over both charges and integrated over the transverse momentum range $0<p_{\text{T}}< 40$ GeV/$c$ and the rapidity range $2.0<y<4.5$ are measured to be $\sigma(pp \rightarrow B^{\pm} X, \sqrt{s} = \text{7 TeV}) = 43.0 \pm 0.2 \pm 2.5 \pm 1.7\mu b,$ $\sigma(pp \rightarrow B^{\pm} X, \sqrt{s} = \text{13 TeV}) = 86.6 \pm 0.5 \pm 5.4 \pm 3.4\mu b,$ where the first uncertainties are statistical, the second are systematic, and the third are due to the limited knowledge of the $B^{\pm}\rightarrow J/\psi K^{\pm}$ branching fraction. The ratio of the cross-section at 13 TeV to that at 7 TeV is determined to be $2.02\pm0.02\text{(stat)}\pm0.12\text{(syst)}$. Differential cross-sections are also reported as functions of $p_{\text{T}}$ and $y$. All results are in agreement with theoretical calculations based on the state-of-art fixed next-to-leading order quantum chromodynamics.
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