Using decays to $\phi$meson pairs, the inclusive production of charmonium states in $b$hadron decays is studied with $pp$ collision data corresponding to an integrated luminosity of $3.0fb^{1}$, collected by the LHCb experiment at centreofmass energies of 7 and 8 TeV. Denoting by $B_C\equiv B(b\to CX)\times B(C\to\phi\phi)$ the inclusive branching fraction of a $b$ hadron to a 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 first uncertainties 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 crosssections as a function of transverse momentum are measured for 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, $B(B_s^0\to\phi\phi\phi)=(2.15\pm0.54\pm0.28\pm0.21_{B})\times 10^{6}$. Here the third uncertainty is due to the branching fraction of the decay $B_s^0\to\phi\phi$, which is used for normalization. No evidence for intermediate resonances is seen. A preferentially transverse $\phi$ polarization is observed. The measurements allow the determination of the ratio of the branching fractions for the $\eta_c(1S)$ decays to $\phi\phi$ and $p\bar{p}$ as $B(\eta_c(1S)\to\phi\phi)/B(\eta_c(1S)\to p\bar{p})=1.79\pm0.14\pm0.32$.
Result of the 2D fit to the $2 ( K^+ K^ )$ invariant mass distribution along with the projections to the $ K ^+ K ^ $ invariant mass axes in the $\eta _ c (1S)$ signal region. 
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Result of the 2D fit to the $2 ( K^+ K^ )$ invariant mass distribution along with the projections to the $ K ^+ K ^ $ invariant mass axes in the $ B ^0_ s $ signal region. 
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Projections to the $ K ^+ K ^ $ invariant mass axes of the 3D fit to the $3 ( K^+ K^ )$ invariant mass distribution in the $ B ^0_ s $ signal region. 
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Distribution of the invariant mass of $\phi \phi$ combinations. The number of candidates in each bin is obtained from the corresponding 2D fit. The peaks corresponding to the $ c \overline c $ resonances are marked on the plot. The signal yields are given in Table 1. 
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Ratio of the $\eta _ c (2S)$ and $\eta _ c (1S)$ inclusive yields $\frac{\mathcal{B} ( b \rightarrow \eta _ c (2S) X ) \times \mathcal{B} ( \eta _ c (2S) \rightarrow \phi \phi )}{\mathcal{B} ( b \rightarrow \eta _ c (1S) X ) \times \mathcal{B} ( \eta _ c (1S) \rightarrow \phi \phi )}$ as a function of the assumed $\eta _ c (2S)$ natural width. Statistical (green band) and total uncertainties are shown separately. The $\eta _ c (2S)$ natural width from Ref. [14] is shown as a vertical solid line; the dashed lines indicate its uncertainty. 
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Differential crosssections normalized to the production crosssection integrated over the studied region, $\sigma^*$, of the (top to bottom) $\eta _ c (1S)$, $\chi _{ c 0}$ , $\chi _{ c 1}$ and $\chi _{ c 2}$ states for the (left) $\sqrt{s} = 7 \mathrm{\,TeV} $ and the (right) $\sqrt{s} = 8 \mathrm{\,TeV} $ data samples. The horizontal and vertical size of the boxes reflect the size of the $p_{\mathrm{ T}}$ bins and the statistical and uncorrelated systematic uncertainties of the differential production crosssections added in quadrature. The exponential functions proportional to $\exp ( \alpha \, p_{\mathrm{ T}} )$ fitted to the integral of the each bin of the distributions are overlaid. 
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Contour plot of $\Gamma_{\eta _ c (1S)}$ and $M_{\eta _ c (1S)}$ using $\eta _ c \rightarrow \phi \phi$ decays. The two magenta curves indicate $\Delta\chi^2 = 1$ and $\Delta\chi^2 = 4$ contours. Only statistical uncertainties are shown. The red cross, black square and blue triangle with error bars indicate the world average [14], the result from Ref. [15], and the result from Ref. [61], respectively. 
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Invariant mass spectrum of the $\phi \phi \phi$ combinations in the region of the $ B ^0_ s $ mass, including the fit function described in the text. 
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Invariant mass spectrum of the $\phi \phi$ combinations in the region of the $ B ^0_ s $ mass, including the fit function described in the text. 
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The invariant mass distribution of each combination of $\phi \phi$ pairs in the $ B ^0_ s \rightarrow \phi \phi \phi$ candidates. The $ B ^0_ s $ candidates are constrained to the known $ B ^0_ s $ mass. A phasespace distribution as obtained from simulation (red histogram) is overlaid. 
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The $\phi$ meson angular distribution for the $ B ^0_ s \rightarrow \phi \phi \phi$ candidates (points with error bars) with the overlaid distribution from the simulation with no polarization (red solid histogram) and two extreme, transverse (green dashed histogram) and longitudinal (blue dotted histogram), polarizations. 
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Animated gif made out of all figures. 
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Signal yields with statistical uncertainties of the fit to the spectrum of the $\phi \phi$ invariant mass. 
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The ratio of charmonium signal yields with respect to the $\eta _ c (1S)$ yield and between pairs of $\chi _{c}$ states. The first uncertainties are statistical and the second systematic. 
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Systematic uncertainties of the charmonium event yield ratios within families and with respect to the $\eta _ c (1S)$ yield. The total uncertainty is the sum in quadrature of the individual contributions. 
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Exponential slope parameter in units of $\mathrm{\,GeV} ^{1}$ from a fit to the $p_{\mathrm{ T}}$ spectra of $\eta _ c (1S)$, $\chi _{ c 0}$ , $\chi _{ c 1}$ and $\chi _{ c 2}$ mesons. 
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Charmonium masses and natural widths in$\, \! \mathrm{\,MeV} $. 
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Charmonium mass differences (in$\, \! \mathrm{\,MeV} $). 
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Created on 16 February 2019.Citation count from INSPIRE on 21 February 2019.