Measurement of the ${\eta _{c}} (1S)$ production cross-section in $p$ $p$ collisions at $\sqrt{s} = 13$ $\, \text {TeV}$

Abstract

Using a data sample corresponding to an integrated luminosity of 2.0 $\, \text {fb} ^{-1}$, collected by the LHCb experiment, the production of the ${\eta _{c}} (1S)$ state in proton–proton collisions at a centre-of-mass energy of $\sqrt{s} =13$ $\, \text {TeV}$ is studied in the rapidity range $2.0< y < 4.5$ and in the transverse momentum range $6.5< p_{\mathrm {T}} < 14.0 \, \text {GeV}$. The cross-section for prompt production of ${\eta _{c}} (1S)$ mesons relative to that of the ${J /\psi }$ meson is measured using the $p$ $\overline{{p}}$ decay mode and is found to be ${\sigma _{{\eta _{c}} (1S)}/\sigma _{{{J /\psi }}} = 1.69 \pm 0.15 \pm 0.10 \pm 0.18}$. The quoted uncertainties are, in order, statistical, systematic and due to uncertainties on the branching fractions of the ${{J /\psi }} \!\rightarrow {p} {\overline{{p}}}$ and ${\eta _{c}} \!\rightarrow {p} {\overline{{p}}}$ decays. The prompt ${\eta _{c}} (1S)$ production cross-section is determined to be ${\sigma _{{\eta _{c}} (1S)} = 1.26 \pm 0.11\pm 0.08\pm 0.14 \,\upmu \text {b}}$, where the last uncertainty includes that on the ${J /\psi }$ meson cross-section. The ratio of the branching fractions of $b$-hadron decays to the ${\eta _{c}} (1S)$ and ${J /\psi }$ states is measured to be ${{\mathcal {B}} _{{b} \!\rightarrow {\eta _{c}} X}/{\mathcal {B}} _{{b} \!\rightarrow {{J /\psi }} X} = 0.48 \pm 0.03 \pm 0.03 \pm 0.05}$, where the last uncertainty is due to those on the branching fractions of the ${{J /\psi }} \!\rightarrow {p} {\overline{{p}}}$ and ${\eta _{c}} \!\rightarrow {p} {\overline{{p}}}$ decays. The difference between the ${J /\psi }$ and ${\eta _{c}} (1S)$ masses is also determined to be $113.0 \pm 0.7 \pm 0.1 \, \text {MeV}$, which is the most precise single measurement of this quantity to date.

1 Introduction

Nonrelativistic quantum chromodynamics (NRQCD) [1] is a powerful framework to describe the production from initial parton scattering (hadroproduction) of charmonium states with quantum numbers $J^{PC} = 1^{--}$, for instance the ${J /\psi }$ meson, over a wide range of transverse momentum, $p_{\mathrm {T}}$, and rapidity, y. Nevertheless, it remains a challenge to provide a comprehensive theoretical description of measurements of the prompt production, comprising the hadroproduction and feed-down from excited resonant states, and the polarisation of ${J /\psi }$ mesons at the Tevatron  [2] and the LHC  [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] collision energies over the entire $p_{\mathrm {T}}$ range.

A factorisation approach together with a heavy-quark spin symmetry assumption allows the simultaneous treatment of ${J /\psi }$ and ${\eta _{c}} (1S)$ (the ground-level charmonium state with $J^{PC}=0^{-+}$ quantum numbers)¹ production observables by imposing relations between their long-distance matrix elements (LDME) [1]. The LHCb collaboration measured the prompt ${\eta _{c}}$ production cross-section in proton–proton collisions at centre-of-mass energies of $\sqrt{s} = 7$ and $8 \, \text {TeV}$ [18] to be below the predictions based on ${J /\psi }$ prompt production data [19,20,21,22,23], which motivated several groups to revisit the theoretical approach [24,25,26, 28,29,30,31]. A study of the ${\eta _{c}}$ prompt production at $\sqrt{s} = 13 \, \text {TeV}$ provides a further important test for theories predicting the ${J /\psi }$ and ${\eta _{c}}$ hadroproduction cross-sections and the ${J /\psi }$ polarisation [32].

The LHCb collaboration has also measured the branching fractions of inclusive $b$-hadron decays to ${\eta _{c}}$ [18] and to $\chi _{{c} 0}$, $\chi _{{c} 1}$ and $\chi _{{c} 2}$ mesons [33]. At the LHC, the $b$-hadron sample comprises a mixture of ${{B} ^+}$, ${B} ^0$, ${B} ^0_{s}$, ${B} _{c} ^+$ mesons and $b$  baryons.² A simultaneous study of the hadroproduction and production in inclusive $b$-hadron decays of the charmonium states with linked LDMEs provides a unique test of basic NRQCD assumptions [34]. Only marginal consistency was found between measurements and theoretical predictions at next-to-leading-order [21, 35] for both prompt production and production in $b$-hadron decays for the ${\eta _{c}}$ and ${J /\psi }$ states.

Using a sample of ${{{{B} ^+}} \!\rightarrow {p} {\overline{{p}}} {{K} ^+} }$ decays, the LHCb collaboration has recently measured [36] the mass difference of the ${J /\psi }$ and ${\eta _{c}}$ states, $\Delta M_{{{J /\psi }}, {\eta _{c}}}$ to be 2.8 standard deviations smaller than the world-average value [37]. A dataset of $b$-hadron decays to the $\eta _{c}$ meson can be analysed to measure the $\Delta M_{{{J /\psi }}, {\eta _{c}}}$ with improved precision.

This paper reports measurements of the $\eta _{c}$ prompt production cross-section and branching fraction of $b$-hadron inclusive decays to the $\eta _{c}$ meson. A dedicated selection of $\eta _{c}$ mesons produced in $b$-hadron decays is developed to perform the most precise measurement of the $\Delta M_{{{J /\psi }}, {\eta _{c}}}$. Both ${J /\psi }$ and $\eta _{c}$ mesons are reconstructed via their decays to $p$ $\overline{{p}}$.

2 The LHCb detector and data sample

The LHCb detector [38, 39] is a single-arm forward spectrometer covering the pseudorapidity range $2<\eta <5$, designed for the study of particles containing $b$ or $c$ quarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the pp interaction region, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about $4{\mathrm {\,Tm}}$, and three stations of silicon-strip detectors and straw drift tubes placed downstream of the magnet. The tracking system provides a measurement of the momentum, $p$, of charged particles with a relative uncertainty that varies from 0.5% at low momentum to 1.0% at 200$\, \text {GeV}$.³ The minimum distance of a track to a $p$ $p$ collision vertex (PV), the impact parameter (IP), is measured with a resolution of $(15+29/p_{\mathrm {T}})\,\upmu \text {m}$, where $p_{\mathrm {T}}$ is given in $\, \text {GeV}$. Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov (RICH) detectors. Photons, electrons and hadrons are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers. The online event selection is performed by a trigger, which consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction.

The analysis is based on $p$ $p$ collision data recorded by the LHCb experiment in 2015 and 2016 at a centre-of-mass energy of $13 \, \text {TeV}$, corresponding to an integrated luminosity of $2.0 \, \text {fb} ^{-1}$. Events enriched in signal decays are selected by the hardware trigger based on the presence of a single deposit of high transverse energy in the calorimeter. The trigger also specifically rejects high-multiplicity events, which produce excessive random combinations of tracks (combinatorial background). The subsequent software trigger selects charged particles with a good track-fit quality and $p_{\mathrm {T}} > 2\, \text {GeV}$. Proton candidates are identified using information from the RICH and tracking detectors. Pairs of oppositely charged proton candidates are required to form a good quality vertex and to have $p_{\mathrm {T}} > 6.5\, \text {GeV}$. The selection follows that of Ref. [18]. The signal selection of both prompt charmonia and charmonia from $b$-hadron decays is largely performed at the trigger level.

For the measurement of the ${\eta _{c}}$ mass, a low-background data sample enriched in ${b} \!\rightarrow {\eta _{c}} X$ decays is used. This sample is obtained using a software-trigger selection based on a multivariate algorithm that requires the presence of two, three or four charged tracks that form a common vertex and are inconsistent with originating from a PV [40, 41]. Precise mass measurements require a momentum-scale calibration. The absolute scale is determined using ${{{B} ^+}} \rightarrow {{J /\psi }} {{K} ^+}$ decays with known particle masses as input [37]. Decays of ${{J /\psi }} \rightarrow {\mu ^+} {\mu ^-}$ are used to cross-calibrate a relative momentum scale between different data-taking periods [42]. The final calibration is checked with a variety of reconstructed quarkonia, ${{B} ^+}$ and ${K} ^0_{\mathrm {S}}$ meson decays. No residual momentum bias is observed within the experimental resolution.

Samples of simulated events are used to model the effects of the detector acceptance and the imposed selection requirements. In the simulation, pp collisions are generated using Pythia  [43, 44] with a specific LHCb configuration [45]. Decays of hadronic particles are described by EvtGen  [46], in which final-state radiation is generated using Photos  [47]. The interaction of the generated particles with the detector material, and its response, are implemented using the Geant4 toolkit [48, 49] as described in Ref. [50]. A simulated sample of ${{J /\psi }} \!\rightarrow {p} {\overline{{p}}} {{\pi } ^0}$ decays is used to study the corresponding background contribution. The $\eta _{c}$ and ${J /\psi }$ decays are generated with uniform phase-space density, and the prompt ${J /\psi }$ mesons are generated without polarisation. Inclusive $b$-hadron decays are modelled using a combination of many exclusive final states based on measurements from the $B$ factories, Tevatron and LHC experiments [46].

3 Analysis technique

In this analysis, $\eta _{c}$ production is studied in a fiducial region of ${6.5<p_{\mathrm {T}} <14.0 \, \text {GeV}}$ and ${2.0<y<4.5}$. The measurement of the differential production cross-section is performed as a function of the transverse momentum relative to that of the ${J /\psi }$ meson [6, 37]. Both the $\eta _{c}$ and the ${J /\psi }$ mesons are reconstructed in the $p$ $\overline{{p}}$ final state. The measured ratio is determined as

$$\begin{aligned} \begin{aligned} \frac{\sigma ^{\text {prompt ({b})}}_{{\eta _{c}}}}{\sigma ^{\text {prompt ({b})}}_{{{J /\psi }}}}&= \frac{N^{\text {prompt ({b})}}_{{\eta _{c}}}}{N^{\text {prompt ({b})}}_{{{J /\psi }}}}\times \frac{\epsilon _{{{J /\psi }}}}{\epsilon _{{\eta _{c}}}}\times \frac{{\mathcal {B}} _{{{J /\psi }} \!\rightarrow {p} {\overline{{p}}} }}{{\mathcal {B}} _{{\eta _{c}} \!\rightarrow {p} {\overline{{p}}} }}, \end{aligned} \end{aligned}$$

(1)

where $\sigma ^{\text {prompt}}_{{\eta _{c}}}$ is the ${\eta _{c}}$ prompt production cross-section and $\sigma ^{{b}}_{{\eta _{c}}}$ is the production cross-section in inclusive $b$-hadron decays; $N^{\text {prompt}}_{{\eta _{c}}}$ and $N^{{b}}_{{\eta _{c}}}$ are the signal yields of $\eta _{c}$ mesons produced promptly and in $b$-hadron decays, respectively. Similar definitions apply for the ${J /\psi }$ yields and cross-sections. The $\frac{\epsilon _{{{J /\psi }}}}{\epsilon _{{\eta _{c}}}}$ is the ratio of total efficiencies to trigger, reconstruct and select ${{J /\psi }} \!\rightarrow {p} {\overline{{p}}}$ and ${\eta _{c}} \!\rightarrow {p} {\overline{{p}}}$ decays, which is found to be the same, within uncertainties, for prompt and $b$-decay charmonia. The ratio of branching fractions of $b$-hadron inclusive decays to $\eta _{c}$ and to ${J /\psi }$ mesons, $\frac{{\mathcal {B}} _{{b} \!\rightarrow {\eta _{c}} X}}{{\mathcal {B}} _{{b} \!\rightarrow {{J /\psi }} X}}=\frac{\sigma ^{{b}}_{{\eta _{c}}}}{\sigma ^{{b}}_{{{J /\psi }}}}$, is defined in the same way as for prompt production. The values of the branching fractions of the $\eta _{c}$ and ${J /\psi }$ decays to $p$ $\overline{{p}}$, ${\mathcal {B}} _{{\eta _{c}} \!\rightarrow {p} {\overline{{p}}} }=(1.50\pm 0.16)\times 10^{-3}$ and ${\mathcal {B}} _{{{J /\psi }} \!\rightarrow {p} {\overline{{p}}} }=(2.120\pm 0.029)\times 10^{-3}$, correspond to the current world-average values [37]. While the branching fraction of $b$-hadron inclusive decays to ${J /\psi }$ meson, ${\mathcal {B}} _{{b} \!\rightarrow {{J /\psi }} X} = (1.16\pm 0.10) \times 10^{-3}$, was measured at LEP  [37], this analysis assumes the same value for the $b$-hadron mixture at LHC.

Since the masses of the $\eta _{c}$ and ${J /\psi }$ states and kinematic distributions in ${{J /\psi }} \!\rightarrow {p} {\overline{{p}}}$ and ${\eta _{c}} \!\rightarrow {p} {\overline{{p}}}$ decays are similar, they have similar reconstruction, trigger and selection efficiencies. Using simulation, the efficiency ratio of the ${{J /\psi }} \!\rightarrow {p} {\overline{{p}}}$ and ${\eta _{c}} \!\rightarrow {p} {\overline{{p}}}$ modes is determined to be $\epsilon _{{{J /\psi }}}/\epsilon _{{\eta _{c}}} = 1.00\pm 0.02$, where the uncertainty is due to the size of the simulation samples. The efficiency ratio is also obtained in bins of $p_{\mathrm {T}}$, with negligible deviation from unity observed. Prompt ${J /\psi }$ mesons are assumed to be unpolarised. A systematic uncertainty is further assigned related to possible non-zero polarisation.

In the baseline analysis, promptly produced charmonium candidates are distinguished from those originating from $b$-hadron decays using the pseudo-proper decaytime

$$\begin{aligned} t_z = \frac{\Delta z \cdot M_{{p} {\overline{{p}}}}}{p_{z}}\, , \end{aligned}$$

(2)

where $\Delta z$ is the distance along the beam axis between the PV with the smallest IP significance of the charmonium candidate and the charmonium decay vertex; $M_{{p} {\overline{{p}}}}$ is the reconstructed charmonium mass; and ${p_{z}}$ is the projection of its momentum along the beam axis. A prompt-enriched sample is selected with candidates satisfying $t_z < 80 \, \text {fs}$, while a $b$-hadron-enriched sample is selected with $t_z > 80 \, \text {fs}$ and an additional requirement that both proton tracks are significantly displaced from any PV. These two samples have a small percentage of wrongly classified candidates. The probability of such cross-feed is estimated using simulation and is used to derive corrected yields. The cross-feed correction on the yield ratio ranges from 1.1 to 2.7% in the prompt-enriched sample and 0.7 to 1.2% in the $b$-hadron-enriched sample, depending on the charmonium $p_{\mathrm {T}}$.

A cross-check of the results, reported in Appendix A, uses an alternative approach analysing the $t_z$ distribution of the selected candidates. The results are in good agreement with the baseline analysis.

4 Fit to the invariant mass

A binned fit to the $p$ $\overline{{p}}$ invariant mass of the prompt-enriched and $b$-hadron-enriched data samples is performed simultaneously in each bin of charmonium $p_{\mathrm {T}}$ in order to extract the ${J /\psi }$ signal yield and the $\eta _{c}$-to-${J /\psi }$ yield ratio. For the $\eta _{c}$ state, the signal shape is modelled by a relativistic Breit–Wigner function convolved with the sum of two Gaussian functions, while the signal shape of the ${J /\psi }$ state is modelled by the sum of two Gaussian functions. In the study of the $\eta _{c}$ production, the mass values $M_{{{J /\psi }}}$ and $\Delta M_{{{J /\psi }},\,{\eta _{c}}}$ are constrained within uncertainties in each $p_{\mathrm {T}}$ bin to the values obtained from a fit to the entire data sample, where they are found to be consistent with the known values [37]. The mass resolutions of charmonium from $b$-hadron decays and from prompt production are assumed to be the same, as confirmed by simulation. The ratio between the widths of the resolution functions for the ${J /\psi }$ and $\eta _{c}$ mesons is fixed from simulation. The only resolution parameter left free to vary in the fit is the width of the narrower Gaussian in the $\eta _{c}$ model. A small $p_{\mathrm {T}}$-dependence of resolution parameters is seen in the simulation and is accounted for in the fit. The natural width of the $\eta _{c}$ meson is fixed to its known value [37]. The combinatorial background is parametrised using an exponential multiplied by a second-order polynomial. A partially reconstructed background of proton–antiproton pairs from the decays of higher mass charmonium states could exhibit structures in the ${p} {\overline{{p}}}$ invariant mass spectrum. The only contribution relevant for this analysis is that from ${{J /\psi }} \!\rightarrow {p} {\overline{{p}}} {{\pi } ^0}$ decays, where the ${\pi } ^0$ meson is not reconstructed. This background produces a broad, non peaking, contribution to the $p$ $\overline{{p}}$ invariant mass below the known $\eta _{c}$ mass. In this region, the ${p} {\overline{{p}}} {{\pi } ^0}$ background is described by a square-root shape, $\sqrt{M_T-M_{{p} {\overline{{p}}}}}$, below the phase-space limit, $M_T$. The yield of this contribution is related to that of the decay ${{{J /\psi }}}\rightarrow {{p} {\overline{{p}}}}$ by means of the ratio of branching fractions ${\mathcal {B}} _{{{J /\psi }} \!\rightarrow {p} {\overline{{p}}} {{\pi } ^0} } / {\mathcal {B}} _{{{J /\psi }} \!\rightarrow {p} {\overline{{p}}} } = 0.56\pm 0.04$ [37] and the ratio of efficiencies $\epsilon _{{{J /\psi }} \!\rightarrow {p} {\overline{{p}}} {{\pi } ^0} }/\epsilon _{{{J /\psi }} \!\rightarrow {p} {\overline{{p}}} } = 0.062\pm 0.002$ for considered $p$ $\overline{{p}}$ invariant mass window.

The $p$ $\overline{{p}}$ invariant mass of selected candidates is shown in Fig. 1. Projections of the simultaneous fit result integrated over the entire $p_{\mathrm {T}}$ range are overlaid. In general, the fit provides a good description of all $M_{{p} {\overline{{p}}}}$ distributions. The charmonium yields in bins of $p_{\mathrm {T}}$ and for the entire data sample are summarised in Table 1. These yields are corrected to take into account the cross-feed probabilities.

Fig. 1

Invariant-mass distribution of the $p$ $\overline{{p}}$ candidates for (top left) prompt-enriched and (top right) $b$-hadron-enriched samples in the entire $p_{\mathrm {T}}$ range, $6.5<p_{\mathrm {T}} <14.0 \, \text {GeV}$. The solid blue lines represent the total fit result. The dashed magenta and dotted red lines show the signal and background components, respectively. Signal distributions with the background components from the fit subtracted are shown on the bottom plots for the two samples

Table 1 Yield of ${J /\psi }$ mesons and the $\eta _{c}$-to-${J /\psi }$ yield ratio for prompt and $b$-hadron decay production, corrected for the cross-feed, in bins of transverse momentum

5 Systematic uncertainties

The systematic uncertainties on the $\eta _{c}$ production corresponding to the signal and background descriptions in the invariant-mass fit are estimated using alternative fit models. Each uncertainty is estimated in $p_{\mathrm {T}}$ bins as a difference between the baseline fit result and the alternative fit result. Generally, bin-to-bin variations of uncorrelated systematic uncertainties addressed in this paragraph are small compared to the statistical uncertainties. These variations are parametrised using a linear $p_{\mathrm {T}}$ dependence to reduce statistical fluctuations. The uncertainty related to a potential $p_{\mathrm {T}}$ dependence of the resolution ratio is evaluated by modelling it using a linear function with the slope constrained by simulation. This systematic effect is relevant for the differential cross-section measurement. The uncertainty corresponding to the combinatorial background description is estimated by using an alternative model using a third-order polynomial function. The uncertainty associated to the ${{J /\psi }} \!\rightarrow {p} {\overline{{p}}} {{\pi } ^0}$ background contribution is estimated by varying the efficiency ratio $\epsilon _{{{J /\psi }} \!\rightarrow {p} {\overline{{p}}} {{\pi } ^0} }/\epsilon _{{{J /\psi }} \!\rightarrow {p} {\overline{{p}}} }$ and the branching fraction ratio ${\mathcal {B}} _{{{J /\psi }} \!\rightarrow {p} {\overline{{p}}} }/{\mathcal {B}} _{{{J /\psi }} \!\rightarrow {p} {\overline{{p}}} {{\pi } ^0} }$ [37] within their uncertainties.

The systematic uncertainty associated with the cross-feed probability is estimated by varying the efficiency of the separation requirements. Separation efficiencies are found to be in good agreement between data and simulation within uncertainties. The uncertainty related to the ratio of $\eta _{c}$ and ${J /\psi }$ efficiencies is estimated by varying its value by the uncertainty corresponding to the simulation sample sizes.

A potential effect due to the invariant-mass resolution modelling is evaluated considering, as an alternative model, the sum of two Crystal Ball functions [51] with symmetric tails on both sides of the peak. The uncertainty on the $\eta _{c}$ natural width is accounted for by the difference in relative yields when using the world average value of $31.9 \pm 0.7$ $\, \text {MeV}$  [37] and the value, $34.0 \pm 1.9 \pm 1.3$ $\, \text {MeV}$, recently measured by LHCb collaboration [36]. Since this uncertainty is correlated among $p_{\mathrm {T}}$ bins, the relative systematic uncertainty obtained from the $p_{\mathrm {T}}$-integrated data sample is taken as an estimate of the relative systematic uncertainty in each bin. This uncertainty is also correlated between $p_{\mathrm {T}}$ bins. Possible nonzero polarisation of prompt ${J /\psi }$ mesons affects their reconstruction efficiency. The ${J /\psi }$ polarisation has not been measured at $\sqrt{s}$ =13$\, \text {TeV}$, although several experiments have measured small polarisation values at lower energy [13, 15, 16]. The associated systematic uncertainty is estimated by weighting the prompt ${J /\psi }$ simulation sample assuming polarisation parameter values $\lambda _{\theta }=\pm 0.1$ [52] in the $p$ $p$ collision frame. This uncertainty is correlated among $p_{\mathrm {T}}$ bins.

Systematic uncertainties on the relative cross-sections of the $\eta _{c}$ production for prompt and $b$-hadron decays are given in Tables 2 and 3. The total systematic uncertainty is calculated as the quadratic sum of the individual sources. The dominant source of uncorrelated systematic uncertainty for the production of $\eta _{c}$ meson for both prompt and from $b$-hadron decays is related to the combinatorial background description. The dominant sources of correlated systematic uncertainty for prompt production are related to the knowledge of the $\eta _{c}$ natural width and the invariant-mass resolution model. The uncertainty on the knowledge of the $\eta _{c}$ natural width is the dominant source of correlated systematic uncertainty for $b$-hadron decay production. Systematic uncertainties are in general smaller than the corresponding statistical uncertainties.

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

Table 3 Relative uncertainties (in %) on the ratio of cross-sections for production in inclusive $b$-hadron decays $\sigma ^{b}_{{\eta _{c}}}/\sigma ^{b}_{{{J /\psi }}}$. Uncertainties on ${\mathcal {B}} _{{\eta _{c}} \!\rightarrow {p} {\overline{{p}}} }$ and ${\mathcal {B}} _{{{J /\psi }} \!\rightarrow {p} {\overline{{p}}} }$ are considered separately and given in the text

Uncertainties on the branching fractions of the ${{J /\psi }} \!\rightarrow {p} {\overline{{p}}}$ and ${\eta _{c}} \!\rightarrow {p} {\overline{{p}}}$ decays are considered separately. They are correlated among $p_{\mathrm {T}}$ bins and amount to about 10%. When deriving the absolute ${\eta _{c}}$ production cross-section, the uncertainty on the ${J /\psi }$ production cross-section [6] is also taken into account.

6 Results and discussion on the $\eta _{c}$ production

Using Eq. 1 and the corrected yields from Table 1, the relative prompt production cross-section in the chosen fiducial region is measured to be

$$\begin{aligned}&\left( \sigma _{{\eta _{c}}}^{\text {prompt}}/\sigma _{{{J /\psi }}}^{\text {prompt}} \right) _{13 \, \text {TeV}}^{6.5< p_{\mathrm {T}}< 14.0 \, \text {GeV},\,2.0<y<4.5} \\&\quad = 1.69 \pm 0.15 \pm 0.10 \pm 0.18. \end{aligned}$$

Here and hereinafter, the quoted uncertainties are statistical, systematic and systematic due to uncertainties on the branching fractions ${\mathcal {B}} _{{{J /\psi }} \!\rightarrow {p} {\overline{{p}}} }$ and ${\mathcal {B}} _{{\eta _{c}} \!\rightarrow {p} {\overline{{p}}} }$, respectively. Using the corresponding cross-section value of $0.749 \pm 0.005 \pm 0.028 \pm 0.037 \,\upmu \text {b}$ for prompt ${J /\psi }$ production [6], the prompt $\eta _{c}$ production cross-section in the chosen fiducial region is derived to be

$$\begin{aligned}&\left( \sigma _{{\eta _{c}}}^{\text {prompt}} \right) _{13 \, \text {TeV}}^{6.5< p_{\mathrm {T}}< 14.0 \, \text {GeV},\,2.0<y<4.5} \\&\quad = 1.26 \pm 0.11\pm 0.08\pm 0.14 \,\upmu \text {b}, \end{aligned}$$

where the last uncertainty includes in addition the uncertainty on the ${J /\psi }$ production cross-section measurement. This is the first measurement of prompt $\eta _{c}$ production cross-section in proton–proton collisions at $\sqrt{s}$ =13$\, \text {TeV}$, and it supports the conclusions of Ref. [18] that suggests an enhanced $\eta _{c}$ production compared to that of the ${J /\psi }$ meson. This measurement is in good agreement with the colour-singlet model prediction of $1.56^{+0.83}_{-0.49}\,^{+0.38}_{-0.17}\,\upmu \text {b}$ [32]. This result leaves limited room for a potential colour-octet contribution, and confirms the theoretical analyses [20,21,22,23] following the $\eta _{c}$ production studies at ${\sqrt{s} =7}$ and ${8 \, \text {TeV}}$ [18].

Using the LHCb measurements of prompt $\eta _{c}$ production at the centre-of-mass energies ${\sqrt{s} =7}$ and ${8 \, \text {TeV}}$ [18], the prompt $\eta _{c}$ production cross-section dependence on the LHC energy is shown in Fig. 2. The ${J /\psi }$ production cross-section from Ref. [6] is also shown for reference. While the individual cross-sections grow with centre-of-mass energy, no evolution of the cross-section ratio is observed.

Fig. 2

The prompt $\eta _{c}$ production cross-section as a function of centre-of-mass energy. (Left) Relative $\eta _{c}$ prompt production cross-section. (Right) Absolute $\eta _{c}$ (black rectangles) and ${J /\psi }$ (blue circles) prompt production cross-sections. The error bars show uncertainties due to statistical, systematic, and to the ${\eta _{c}} \!\rightarrow {p} {\overline{{p}}}$ and ${{J /\psi }} \!\rightarrow {p} {\overline{{p}}}$ branching fractions and ${J /\psi }$ production cross-section

The relative $\eta _{c}$ inclusive branching fraction from $b$-hadron decays is measured to be

$$\begin{aligned} {\mathcal {B}} _{{b} \!\rightarrow {\eta _{c}} X}/{\mathcal {B}} _{{b} \!\rightarrow {{J /\psi }} X}&= 0.48 \pm 0.03 \pm 0.03 \pm 0.05, \end{aligned}$$

which combined with ${\mathcal {B}} _{{b} \!\rightarrow {{J /\psi }} X}=1.16\pm 0.10 \%$ [37] gives

$$\begin{aligned} {\mathcal {B}} _{{b} \!\rightarrow {\eta _{c}} X}&= (5.51 \pm 0.32 \pm 0.29 \pm 0.77)\times 10^{-3}. \end{aligned}$$

The last uncertainty includes the uncertainty on ${\mathcal {B}} _{{b} \!\rightarrow {{J /\psi }} X}$. This result is the most precise measurement of the inclusive ${b} \!\rightarrow {\eta _{c}} X$ branching fraction to date and is in good agreement with the previous LHCb measurement from Ref. [18]. The measurement is limited by the knowledge of the branching fractions ${\mathcal {B}} _{{\eta _{c}} \!\rightarrow {p} {\overline{{p}}} }$ and ${\mathcal {B}} _{{b} \!\rightarrow {{J /\psi }} X}$.

Numerical results of the measurements of $p_{\mathrm {T}}$-differential $\eta _{c}$ production are given in Appendix B. The relative $\eta _{c}$ to ${J /\psi }$ $p_{\mathrm {T}}$-differential cross-sections for prompt and $b$-hadron decay production are compatible to those measured at ${\sqrt{s} =7}$ and ${8 \, \text {TeV}}$ [18] and are shown in Fig. 3. This is the first $p_{\mathrm {T}}$-differential cross-section measurement of $\eta _{c}$ prompt production at ${\sqrt{s} =13 \, \text {TeV}}$. The $p_{\mathrm {T}}$ dependence of the prompt cross-section ratio is found to be linear with a slope of $0.22\pm 0.11 \, \text {GeV} ^{-1}$. While the integrated cross-section is in good agreement with the colour-singlet model prediction [32], a hint of a difference between the ${J /\psi }$ and $\eta _{c}$ slopes motivates the extension of the measurement to larger $p_{\mathrm {T}}$ values. A larger measured slope with respect to the prediction from Ref. [32] would indicate a possible colour-octet contribution.

Fig. 3

Relative $\eta _{c}$ to ${J /\psi }$ differential production cross-sections for (left) prompt production and (right) production in $b$-hadron inclusive decays. The uncertainties are statistical, systematic, and due to the ${\eta _{c}} \!\rightarrow {p} {\overline{{p}}}$ and ${{J /\psi }} \!\rightarrow {p} {\overline{{p}}}$ branching fractions, respectively. For the relative prompt production cross-section, the result of a fit with a linear function is overlaid

The absolute $\eta _{c}$ and ${J /\psi }$ differential production cross-sections are shown in Fig. 4. The exponential slopes for the $\eta _{c}$ and ${J /\psi }$ prompt differential cross-sections are determined from the fit to data points to be $0.41\pm 0.07 \, \text {GeV} ^{-1}$ and $0.57\pm 0.01 \, \text {GeV} ^{-1}$, respectively.

Fig. 4

Differential production cross-sections of $\eta _{c}$ (red rectangles) and ${J /\psi }$ (blue circles) for (left) prompt production and (right) production in $b$-hadron inclusive decays. The uncertainties for $\eta _{c}$ production are statistical, systematic, and due to the ${\eta _{c}} \!\rightarrow {p} {\overline{{p}}}$ and ${{J /\psi }} \!\rightarrow {p} {\overline{{p}}}$ branching fractions and ${J /\psi }$ production cross-section. For the prompt production cross-sections, the results of fits with an exponential function are overlaid. The $p_{\mathrm {T}}$ values of the data points correspond to the average values of the fit function over the bins

7 Measurement of the $\pmb {{{J /\psi }}}$–$\pmb {{\eta _{c}}}$ mass difference

While the prompt ${\eta _{c}}$ production measurement requires stringent selection criteria at the trigger level to compete with the challenging background conditions, charmonia produced in $b$-hadron decays are reconstructed in an environment with a controlled background level and are more suitable for a mass measurement. For this reason, a looser selection is applied for the entire data sample to measure the ${\eta _{c}}$ mass relative to the well-known ${J /\psi }$ mass.

Proton and antiproton candidates are required to have good track-fit quality, to be incompatible with originating from any PV, and to have a transverse momentum greater than 1.0$\, \text {GeV}$. The proton–antiproton system is required to have a vertex with a good fit quality, a large significance, $\chi ^2 _{\text {FD}}>81$, of the distance between this vertex and any PV, and to have a transverse momentum greater than $5.5 \, \text {GeV}$. The contamination of the selected sample from ${J /\psi }$ and $\eta _{c}$ prompt production is estimated to be below $0.1 \%$.

The mass difference $\Delta M _{{{J /\psi }}, \, {\eta _{c}}}$ is extracted from an extended maximum-likelihood fit to the $M_{{p} {\overline{{p}}}}$ distribution. The fit provides a good description of the $p$ $\overline{{p}}$ invariant-mass distribution (Fig. 5) yielding

$$\begin{aligned} \Delta M _{{{J /\psi }}, \, {\eta _{c}}} = 113.0\pm 0.7\pm 0.1 \, \text {MeV}, \end{aligned}$$

where the uncertainties are statistical and systematic.

Fig. 5

Distribution of the ${p} {\overline{{p}}}$ invariant mass. The solid line represents the fit result, while the dashed line shows the $\eta _{c}$ and ${J /\psi }$ signal contributions

The majority of the sources of systematic uncertainty are common to the production measurement. The systematic uncertainty related to the momentum-scale calibration is estimated by comparing the fit result with and without the calibration applied. The total systematic uncertainty is calculated as the quadratic sum of the individual contributions (Table 4). The dominant source of systematic uncertainty is related to the resolution model and its $p_{\mathrm {T}}$ dependence.

Table 4 Systematic uncertainties on the measurement of the ${J /\psi }$–${\eta _{c}}$ mass difference

As a cross-check, the invariant-mass fit is performed simultaneously in seven bins of charmonium transverse momentum to take into account a possible dependence of the resolution on charmonium $p_{\mathrm {T}}$. The value obtained for the mass difference is consistent with the baseline result.

This measurement is currently statistically limited and can be improved with larger data samples. It represents the most precise measurement from a single experiment to date. The result is in good agreement with the PDG value [37], the recent BES III result [53], the latest BaBar measurement [54] and LHCb measurements [18, 33, 36].

8 Summary

Using data corresponding to an integrated luminosity of 2.0$\, \text {fb} ^{-1}$, the prompt ${\eta _{c}}$ production cross-section at a centre-of-mass energy of $\sqrt{s} =13$ $\, \text {TeV}$ is measured for the first time. The ratio of the prompt production rates of the ${\eta _{c}}$ and ${J /\psi }$ states in the fiducial region ${6.5< p_{\mathrm {T}} < 14.0 \, \text {GeV}}$ and ${2.0<y<4.5}$ is measured to be

$$\begin{aligned}&\left( \sigma _{{\eta _{c}}}^{\text {prompt}}/\sigma _{{{J /\psi }}}^{\text {prompt}} \right) _{13 \, \text {TeV}}^{6.5< p_{\mathrm {T}}< 14.0 \, \text {GeV},\,2.0<y<4.5}\\&\quad = 1.69 \pm 0.15 \pm 0.10 \pm 0.18, \end{aligned}$$

where the quoted uncertainties are, in order, statistical, systematic and systematic due to uncertainties on the branching fractions, ${\mathcal {B}} _{{{J /\psi }} \!\rightarrow {p} {\overline{{p}}} }$ and ${\mathcal {B}} _{{\eta _{c}} \!\rightarrow {p} {\overline{{p}}} }$.

Using the prompt ${J /\psi }$ production cross-section measurement at $\sqrt{s} =13 \, \text {TeV}$ [6], the prompt ${\eta _{c}}$ production cross-section in the chosen fiducial region is derived to be

$$\begin{aligned}&\left( \sigma _{{\eta _{c}}}^{\text {prompt}} \right) _{13 \, \text {TeV}}^{6.5< p_{\mathrm {T}}< 14.0 \, \text {GeV},\,2.0<y<4.5}\\&\quad = 1.26 \pm 0.11\pm 0.08\pm 0.14 \,\upmu \text {b}, \end{aligned}$$

where the last uncertainty includes in addition the uncertainty of the ${J /\psi }$ production cross-section measurement. The result is in good agreement with the colour-singlet model prediction [32]. Contrary to NRQCD expectations, a steeper $p_{\mathrm {T}}$ dependence of the ${J /\psi }$ cross-section compared to that of the $\eta _{c}$ is preferred.

The relative ${\eta _{c}}$ inclusive branching fraction from $b$-hadron decays is measured to be

$$\begin{aligned} {\mathcal {B}} _{{b} \!\rightarrow {\eta _{c}} X}/{\mathcal {B}} _{{b} \!\rightarrow {{J /\psi }} X}&= 0.48 \pm 0.03 \pm 0.03 \pm 0.05. \end{aligned}$$

Using ${\mathcal {B}} _{{b} \!\rightarrow {{J /\psi }} X}$ [37] the absolute $\eta _{c}$ inclusive branching fraction is obtained to be

$$\begin{aligned} {\mathcal {B}} _{{b} \!\rightarrow {\eta _{c}} X}&= (5.51 \pm 0.32 \pm 0.29 \pm 0.77)\times 10^{-3}, \end{aligned}$$

where the last uncertainty includes in addition the uncertainty on ${\mathcal {B}} _{{b} \!\rightarrow {{J /\psi }} X}$. This result is consistent with the previous LHCb measurement [18]. Compatible results are obtained with an alternative analysis technique.

The ${J /\psi }$–${\eta _{c}}$ mass difference is measured using an enlarged data sample of ${b} \!\rightarrow {\eta _{c}} X$ decays. The measurement,

$$\begin{aligned} \Delta M_{{{J /\psi }}, \, {\eta _{c}}} = 113.0 \pm 0.7 \pm 0.1 \, \text {MeV}, \end{aligned}$$

is compatible with both the result of Refs. [36] and [37]. It is the most precise ${\eta _{c}}$ mass determination to date.

Appendices

Appendices

Alternative analysis based on $t_z$ distributions

A cross-check of the results is performed using an alternative approach via a two-step procedure. Signal yields in bins of $p_{\mathrm {T}}$ and $t_z$ are obtained from a simultaneous fit to the corresponding $p$ $\overline{{p}}$ invariant mass distributions of candidates in the bin. Prompt and $b$-decay charmonium contributions are then determined using a simultaneous $\chi ^2$ fit to the resulting $t_z$ distributions in $p_{\mathrm {T}}$ bins, similarly to Ref. [6].

Projections of the simultaneous fit for the entire $p_{\mathrm {T}}$-range are shown in Fig. 6 for illustration purposes.

Fig. 6

Invariant mass distributions of the $p$ $\overline{{p}}$ candidates for seven bins of $t_z$ for the $p_{\mathrm {T}}$-integrated sample $6.5<p_{\mathrm {T}} <14.0 \, \text {GeV}$. The solid blue lines represent the total fit result. Dashed magenta and dotted red lines show the signal and background components, respectively. The distributions with the background component of the fit subtracted are shown below

In the alternative analysis discussed in this appendix, the $p$ $\overline{{p}}$ invariant-mass fit is performed simultaneously in 28 two-dimensional bins of $p_{\mathrm {T}}$ and $t_z$ with the model described in Sect. 4. This model is modified to correct for systematic mass shifts as a function of $t_z$. The corrections are derived from simulation, where the same behaviour is observed.⁴

The mass fits result in four $t_z$ distributions, each corresponding to a $p_{\mathrm {T}}$ bin. Promptly produced charmonium is distinguished from that produced in $b$-hadron decays by performing a simultaneous $\chi ^2$ fit to the four $t_z$ distributions. This fit method does not use the bin centre for the value of $t_z$, but rather the average value of the fit function in the bin. The model to describe the $t_z$ distribution comprises contributions due to prompt charmonia, due to charmonia from $b$-hadron decays and a contribution due to candidates with a wrongly associated PV. The prompt charmonium component is parametrised with a function to account only for resolution, while the component related to charmonia produced in inclusive $b$-hadron decays is parametrised by an exponential decay function convolved with the same resolution function. The exponential slope of the decay function, $\tau _b$, is allowed to vary over $p_{\mathrm {T}}$ according to simulation. The resolution is described by the sum of two Gaussian functions, with the width of the narrow Gaussian component a free fit parameter and the other parameters fixed from the simulation. The $p_{\mathrm {T}}$ dependence of the resolution, as obtained from simulation, is taken into account in the $t_z$-fit to data. The contribution due to candidates associated to a wrong PV are described in the same way as in Ref. [6]. Results of the simultaneous fit to $t_z$ for the entire $p_{\mathrm {T}}$ range are shown in Fig. 7.

Fig. 7

Distributions of $t_z$ for (left) ${J /\psi }$ and (right) ${\eta _{c}}$ mesons for the entire $p_{\mathrm {T}}$ range ${6.5<p_{\mathrm {T}} <14.0 \, \text {GeV}}$ and the result of the simultaneous $\chi ^2$ fit. The solid blue lines show the fit function, the red dashed areas show the prompt components, the green dotted areas show the contributions from inclusive $b$-hadron decays, the grey lines show contributions from the events with wrongly associated PV. The $t_z$ values of the data points correspond to the average values of the fit function over the bins

In addition to the sources of systematic uncertainty discussed in Sect. 5 for the baseline analysis, contributions from the signal description in the fit to $t_z$ and from corrections of the invariant-mass peak positions in $t_z$ bins are considered. This approach is free from the systematic uncertainty associated with the cross-feed effect. The dominant sources of systematic uncertainties are the same as for the baseline analysis.

The relative differential cross-sections of $\eta _{c}$ production obtained with this approach are shown in Fig. 8, where they are compared with those from the baseline approach. The two measurements are strongly correlated. The difference between the results obtained with the two techniques varies in $p_{\mathrm {T}}$ bins between factors of 0.5 and 1.5 of the estimated uncorrelated uncertainty.

Fig. 8

Relative $\eta _{c}$ to ${J /\psi }$ differential production cross-sections for (left) prompt production and (right) production in $b$-hadron inclusive decays obtained using the alternative technique (points) and the baseline technique (red boxes). The uncertainties shown are statistical, systematic, and the uncertainty due to the ${\eta _{c}} \!\rightarrow {p} {\overline{{p}}}$ and ${{J /\psi }} \!\rightarrow {p} {\overline{{p}}}$ branching fractions

Tables of $p_{T}$-differential $\eta _{c}$ production cross-sections

1.1 Prompt production of $\eta _{c}$ mesons

The results on relative $p_{\mathrm {T}}$-differential $\eta _{c}$ prompt production are shown in Table 5. Here and in the following tables, the first uncertainty is statistical, the second is the uncorrelated systematic uncertainty, the third is the systematic uncertainty that is correlated among $p_{\mathrm {T}}$ bins, and the last one is related to the ${\mathcal {B}} _{{{J /\psi }} \!\rightarrow {p} {\overline{{p}}} }$ and ${\mathcal {B}} _{{\eta _{c}} \!\rightarrow {p} {\overline{{p}}} }$ branching fractions. The results on $p_{\mathrm {T}}$-differential $\eta _{c}$ prompt production cross-section are shown in Table 6. Here, the last uncertainty includes the uncertainty on the ${J /\psi }$ production cross-section.

Table 5 Relative $p_{\mathrm {T}}$-differential $\eta _{c}$ prompt production cross-section

Table 6 Differential $\eta _{c}$ prompt production cross-section

1.2 Production of $\eta _{c}$ mesons from b-hadron decays

The results on relative $p_{\mathrm {T}}$-differential $\eta _{c}$ production in inclusive $b$-hadron decays are shown in Table 7. The results on $p_{\mathrm {T}}$-differential $\eta _{c}$ production cross-section in inclusive $b$-hadron decays are shown in Table 8. As above, the last uncertainty includes the uncertainty on the ${J /\psi }$ production cross-section.

Table 7 Relative $p_{\mathrm {T}}$-differential $\eta _{c}$ production cross-section in inclusive $b$-hadron decays

Table 8 The $p_{\mathrm {T}}$-differential $\eta _{c}$ production cross-section in inclusive $b$-hadron decays