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Using decays to $\phi$-meson pairs, the inclusive production of charmoniumstates in $b$-hadron decays is studied with $pp$ collision data correspondingto an integrated luminosity of $3.0fb^{-1}$, collected by the LHCb experimentat centre-of-mass energies of 7 and 8 TeV. Denoting by $B_C\equiv B(b\toCX)\times B(C\to\phi\phi)$ the inclusive branching fraction of a $b$ hadron toa charmonium state $C$ that decays into a pair of $\phi$ mesons, ratios$R^{C1}_{C2}\equiv B_{C1}/B_{C2}$ are determined as$R^{\chi_{c0}}_{\eta_c(1S)}=0.147\pm0.023\pm0.011$, $R^{\chi_{c1}}_{\eta_c(1S)}=0.073\pm0.016\pm0.006$, $R^{\chi_{c2}}_{\eta_c(1S)}=0.081\pm0.013\pm0.005$, $R^{\chi_{c1}}_{\chi_{c0}}=0.50\pm0.11\pm0.01$,$R^{\chi_{c2}}_{\chi_{c0}}=0.56\pm0.10\pm0.01$ and $R^{\eta_c(2S)}_{\eta_c(1S)}=0.040\pm0.011\pm0.004$. Here and below the firstuncertainties are statistical and the second systematic. Upper limits at $90\%$confidence level for the inclusive production of $X(3872)$, $X(3915)$ and$\chi_{c2}(2P)$ states are obtained as $R^{X(3872)}_{\chi_{c1}}<0.34$,$R^{X(3915)}_{\chi_{c0}}<0.12$ and $R^{\chi_{c2}(2P)}_{\chi_{c2}}<0.16$.Differential cross-sections as a function of transverse momentum are measuredfor the $\eta_c(1S)$ and $\chi_c$ states. The branching fraction of the decay$B_s^0\rightarrow\phi\phi\phi$ is measured for the first time,$B(B_s^0\to\phi\phi\phi)=(2.15\pm0.54\pm0.28\pm0.21_{B})\times 10^{-6}$. Herethe third uncertainty is due to the branching fraction of the decay$B_s^0\to\phi\phi$, which is used for normalization. No evidence forintermediate resonances is seen. A preferentially transverse $\phi$polarization is observed. The measurements allow the determination of the ratioof the branching fractions for the $\eta_c(1S)$ decays to $\phi\phi$ and$p\bar{p}$ as $B(\eta_c(1S)\to\phi\phi)/B(\eta_c(1S)\top\bar{p})=1.79\pm0.14\pm0.32$.

Study of charmonium production in $b$-hadron decays and first evidence for the decay $B_s^0 \rightarrow \phi \phi \phi$