  Charmonium production cross-section in proton-proton collisions is measured
at the centre-of-mass energy $\sqrt{s}=13\,TeV$ using decays to $p\bar{p}$
final state. The study is performed using a data sample corresponding to an
integrated luminosity of $2.2\,{fb}^{-1}$ collected in 2018 with the $LHCb$
detector. The production cross-section of the $\eta_c$ meson is measured in a
rapidity range of $2.0 < y < 4.0$ and in a transverse momentum range of $5.0 <
p_{T} < 20.0\,{GeV/\it{c}}$, which is extended compared with previous $LHCb$
analyses. The differential cross-section is measured in bins of $p_{T}$ and,
for the first time, of $y$. Upper limits, at 90% and 95% confidence levels, on
the $\eta_c(2S)$ and $h_c(1P)$ prompt production cross-sections are determined
for the first time.
