A study is presented of central exclusive production of $Upsilon(nS)$ states, where the $Upsilon(nS)$ resonances decay to the $mu^+mu^-$ final state, using $pp$ collision data recorded by the LHCb experiment. The cross-section is measured in the rapidity range $2<y(Upsilon)<4.5$ where the muons are reconstructed in the pseudorapidity range $2<eta(mu^pm)<4.5$. The data sample corresponds to an integrated luminosity of 2.9 fb$^{-1}$ and was collected at centre-of-mass energies of $7$ TeV and $8$ TeV. The measured $Upsilon(1S)$ and $Upsilon(2S)$ production cross-sections are begin{eqnarray} sigma(pp to pUpsilon(1S)p) &=& 9.0 pm 2.1 pm 1.7textrm{ pb and} onumber sigma(pp to pUpsilon(2S)p) &=& 1.3 pm 0.8 pm 0.3textrm{ pb}, onumber end{eqnarray} where the first uncertainties are statistical and the second are systematic. The $Upsilon(1S)$ cross-section is also measured as a function of rapidity and is found to be in good agreement with Standard Model predictions. An upper limit is set at 3.4 pb at the 95% confidence level for the exclusive $Upsilon(3S)$ production cross-section, including possible contamination from $chi_b(3P)toUpsilon(3S)gamma$ decays.
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