A search is performed for the central exclusive production of pairs of charmonia produced in proton-proton collisions. Using data corresponding to an integrated luminosity of $3{rm fb}^{-1}$ collected at centre-of-mass energies of 7 and 8 TeV, $J/psi J/psi$ and $J/psipsi(2S)$ pairs are observed, which have been produced in the absence of any other activity inside the LHCb acceptance that is sensitive to charged particles in the pseudorapidity ranges $(-3.5,-1.5)$ and $(1.5,5.0)$. Searches are also performed for pairs of P-wave charmonia and limits are set on their production. The cross-sections for these processes, where the dimeson system has a rapidity between 2.0 and 4.5, are measured to be $$ begin{array}{rl} sigma^{J/psi J/psi} &= 58pm10{(rm stat)} pm 6{(rm syst)} {rm pb} , sigma^{J/psipsi(2S)} &= 63 ^{+27}_{-18}{(rm stat)}pm 10{(rm syst)} {rm pb} , sigma^{psi(2S)psi(2S)} &< 237 {rm pb}, sigma^{chi_{c0}chi_{c0}} &< 69 {rm nb}, sigma^{chi_{c1}chi_{c1}} &< 45 {rm pb}, sigma^{chi_{c2}chi_{c2}} &< 141 {rm pb}, end{array} $$ where the upper limits are set at the 90% confidence level. The measured $J/psi J/psi$ and $J/psipsi(2S)$ cross-sections are consistent with theoretical expectations.
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