Distribution of the $p\bar{p}$ mass for (left) prompt and (right) $b$-decays samples. The results of the fits using the configuration to study the $\eta_c(2S)$ prompt production are overlaid. Red arrows show invariant mass of charmonium states in the following order: $\eta_c$, $J/\psi$, $\chi_{c0}$, $\chi_{c1}$, $h_c(1P)$, $\chi_{c2}$, $\eta_c(2S)$ and $\psi(2S)$ states. The lower panel shows the data and the fit model with the background subtracted.

The log-likelihood profile plots for the (black dashed line) baseline fit and alternative fits including: (red dashed line) alternative mass resolution model, (green and blue dashed lines) alternative models of combinatorial background, (orange dashed line) alternative resolution ratio, (magenta dashed line) alternative contribution from partially reconstructed $J/\psi\rightarrow p\bar{p}\pi^{0}$ decay, and (cyan dashed line) accounting for the nonzero $J/\psi$ polarisation. The black solid line represents the log-likelihood profile obtained using the discrete profiling method \cite{Dauncey:2014xga}. The solid red and green lines show Bayesian upper limits on (left) $\eta_c(2S)$ and (right) relative $\eta_c(2S)$ to $J/\psi$ prompt-production cross-section at 90% and 95% CL, respectively.

The log-likelihood profile plots for the (black dashed line) baseline fit and alternative fits including: (red dashed line) alternative mass resolution model, (green and blue dashed lines) alternative models of combinatorial background, (orange dashed line) alternative resolution ratio, (magenta dashed line) alternative contribution from partially reconstructed $J/\psi\rightarrow p\bar{p}\pi^{0}$ decay, and (cyan dashed line) accounting for the nonzero $J/\psi$ polarisation. The black solid line represents the log-likelihood profile obtained using the discrete profiling method \cite{Dauncey:2014xga}. The solid red and green lines show Bayesian upper limits on (left) $h_c(1P)$ and (right) relative $h_c(1P)$ to $J/\psi$ prompt-production cross-section at 90% and 95% CL, respectively.

Left: measured $\eta_c$ to $J/\psi$ cross-section ratio at different centre-of-mass energies. Right: measured $\eta_c$ and $J/\psi$ \cite{LHCb-PAPER-2013-008,LHCb-PAPER-2013-016,LHCb-PAPER-2015-037} prompt production cross-sections. The red points show the result from the current analysis. The error bars show following uncertainties: statistical, systematic, and due to the $J/\psi\rightarrow p\bar{p}$ and $\eta _ c \rightarrow p\bar{p}$ branching fractions (and $J/\psi$ production cross-section for the absolute production cross-section).

Measured $\eta_c$ to $J/\psi$ prompt-production cross-section ratio in intervals of (left) $p_{\rm{T}}$ and (right) $y$. The error bars show following uncertainties: statistical, systematic, and due to the branching fractions $\mathcal{B} _{J/\psi\rightarrow p\bar{p}}$ and $\mathcal{B} _{\eta _ c \rightarrow p\bar{p}}$ uncertainties. The result of the fit with a linear function for $p_{\rm{T}}$-differential cross-section is overlaid.

Measurement of the differential prompt production cross-section of the (red) $\eta_c$ and (blue) $J/\psi$ mesons \cite{LHCb-PAPER-2015-037} in bins of (left) $p_{\rm{T}}$- and (right) $y$. The error bars show following uncertainties for $\eta_c$ production: statistical, systematic, and the uncertainty due to the $J/\psi\rightarrow p\bar{p}$ and $\eta _ c \rightarrow p\bar{p}$ branching fractions and $J/\psi$ production cross-section. The results of the fits with exponential functions are overlaid.

Measurement of the (left) absolute $\eta_c$ and (right) relative $\eta_c$ to $J/\psi$ $p_{\rm{T}}$-differential prompt-production cross-section. Magenta boxes represent NLO NRQCD CS prediction; green boxes show the sum of NLO NRQCD CS and CO predictions. Blue lines represent the modified NRQCD prediction \cite{Biswal:2023mwu}.

Animated gif made out of all figures.

Treatment of the parameters in the mass fits for the different studies. Parameters are fixed or constrained to their known values \cite{PDG2022}.

Results of the simultaneous fit to the $ p \overline p $ mass distributions for combined prompt and $ b $ -decays data sample. Yields are restricted to be positive with symmetric uncertainties presented. The uncertainties are statistical only.

Ratios of the observed $\eta _ c $ to $ { J \mskip -3mu/\mskip -2mu\psi }$ yields corrected for cross-contamination for prompt production and production in $ b $ -hadron decays in bins of transverse momentum $ p_{\mathrm{T}}$ . The uncertainties are statistical.

Ratios of the observed $\eta _ c $ to $ { J \mskip -3mu/\mskip -2mu\psi }$ yields mesons corrected for cross-contamination for prompt production and production in $ b $ -hadron decays in bins of rapidity $ y $ . The uncertainties are statistical.

Relative uncertainties (in %) on the ratio of prompt cross-sections $\sigma^{p}_{\eta _ c }/\sigma^{p}_{ { J \mskip -3mu/\mskip -2mu\psi } }$. Uncertainties on $\mathcal{B} _{\eta _ c \rightarrow p \overline p }$ and $\mathcal{B} _{ { J \mskip -3mu/\mskip -2mu\psi } \rightarrow p \overline p }$ measurements are considered separately and given in the text.

Upper limits at 90% and 95% CL for the $\eta _ c {(2S)}$ and $ h _ c {(1P)}$ prompt production cross-sections and their ratios to that of the $ { J \mskip -3mu/\mskip -2mu\psi }$ meson, extracted using a Bayesian approach.

Upper limits (UL) at 95% CL for $\eta _ c {(2S)}$ relative to $ { J \mskip -3mu/\mskip -2mu\psi }$ and $\eta _ c {(2S)}$ absolute prompt production cross-section, and its comparison with NLO NRQCD theoretical prediction (Ref. \cite{Lansberg:2017ozx,Shao:2014yta,Gong:2012ug,Bodwin:2015iua}).

Branching fractions of $\chi _{ c J}$ production in inclusive $ b $ -hadron decays. The quoted uncertainties are statistical, systematic and the related to the $\mathcal{B} _{\chi _{ c J} \rightarrow p \overline p }$, $\mathcal{B} _{ { J \mskip -3mu/\mskip -2mu\psi } \rightarrow p \overline p }$, and $\mathcal{B} _{ b \rightarrow { J \mskip -3mu/\mskip -2mu\psi } X }$ uncertainties. The values are compared with results from analysis of charmonia production using decays to $\phi\phi$ \cite{LHCb-PAPER-2017-007} and the world average values \cite{PDG2022}.

Relative uncertainties (in %) on the ratio of prompt cross-sections $\sigma_{\eta _ c }/\sigma_{ { J \mskip -3mu/\mskip -2mu\psi } }$ in bins of the $\eta _ c $ transverse momentum, $ p_{\mathrm{T}}$ . Uncertainties on $\mathcal{B} _{\eta _ c \rightarrow p \overline p }$ and $\mathcal{B} _{ { J \mskip -3mu/\mskip -2mu\psi } \rightarrow p \overline p }$ are considered separately and given in the text.

Relative uncertainties (in %) on the ratio of prompt cross-sections $\sigma_{\eta _ c }/\sigma_{ { J \mskip -3mu/\mskip -2mu\psi } }$ in bins of the $\eta _ c $ rapidity, $ y $ . Uncertainties on $\mathcal{B} _{\eta _ c \rightarrow p \overline p }$ and $\mathcal{B} _{ { J \mskip -3mu/\mskip -2mu\psi } \rightarrow p \overline p }$ are considered separately and given in the text.

Relative $ p_{\mathrm{T}}$ -differential $\eta _ c $ prompt production cross-section.

Relative $ y $ -differential $\eta _ c $ prompt production cross-section.

Absolute $ p_{\mathrm{T}}$ -differential $\eta _ c $ prompt production cross-section.

Absolute $ y $ -differential $\eta _ c $ prompt production cross-section.