The first study of $J/\psi\phi$ production in diffractive processes in proton-proton collisions is presented. The study is based on an LHCb dataset recorded at centre-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 5 fb$^{-1}$. The data disfavour a nonresonant $J/\psi\phi$ production but are consistent with a resonant model including several resonant states observed previously only in $B^+ \to J/\psi\phi K^+$ decays. The $\chi_{c0}(4500)$ state is observed with a significance over $5\sigma$ and the $\chi_{c1}(4274)$ is confirmed with a significance of more than $4\sigma$.
Total $J/\psi(\to \mu^+ \mu^-)\phi(\to K^+ K^-)$ diffractive production cross-section, multiplied by $\mathcal{B}(J/\psi \to \mu^+ \mu^-)$ and $\mathcal{B}(\phi \to K^+ K^-)$ branching ratios.
$\chi_{c1}(4140) \to J/\psi(\to \mu^+ \mu^-)\phi(\to K^+ K^-)$ diffractive production cross-section, multiplied by $\mathcal{B}(J/\psi \to \mu^+ \mu^-)$ and $\mathcal{B}(\phi \to K^+ K^-)$ branching ratios.
$\chi_{c1}(4274) \to J/\psi(\to \mu^+ \mu^-)\phi(\to K^+ K^-)$ diffractive production cross-section, multiplied by $\mathcal{B}(J/\psi \to \mu^+ \mu^-)$ and $\mathcal{B}(\phi \to K^+ K^-)$ branching ratios.
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