Measurement of the $\eta_c(1S)$ production cross-section in $pp$ collisions at $\sqrt{s} = 13$ TeV

Using a data sample corresponding to an integrated luminosity of $2.0\,fb^{-1}$, 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 ${6.5<p_{T}<14.0\text{ GeV}}$. The cross-section for prompt production of $\eta_c(1S)$ mesons relative to that of the $J/\psi$ meson is measured using the ${p\bar{p}}$ 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 ${J/\psi\to p \bar{p}}$ and ${\eta_c\to p \bar{p}}$ decays. The prompt $\eta_c(1S)$ production cross-section is determined to be ${\sigma_{\eta_c(1S)} = 1.26 \pm 0.11\pm 0.08 \pm 0.14 \,\mu b}$, 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 ${\mathcal{B}_{b\to\eta_c X}/\mathcal{B}_{b\to J/\psi X} = 0.48 \pm 0.03 \pm 0.03 \pm 0.05}$, where the last uncertainty is due to those on the branching fractions of the ${J/\psi \to p \bar{p}}$ and ${\eta_c\to p \bar{p}}$ 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.

A study of the η c prompt production at √ s = 13 TeV provides a further important test for theories predicting the J/ψ and η c hadroproduction cross-sections and the J/ψ polarisation [31].
The LHCb collaboration has also measured the branching fractions of inclusive b-hadron decays to η c [18] and to χ c0 , χ c1 and χ c2 mesons [32]. 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 [33]. Only marginal consistency was found between measurements and theoretical predictions at next-to-leading-order [21,34] for both prompt production and production in b-hadron decays for the η c and J/ψ states.
Using a sample of B + → ppK + decays, the LHCb collaboration has recently measured [35] the mass difference of the J/ψ and η c states, ∆M J/ψ ,ηc to be 2.9 standard deviations smaller than the world-average value [36]. A dataset of b-hadron decays to the η c meson can be analysed to measure the ∆M J/ψ ,ηc with improved precision.
This paper reports measurements of the η c prompt production cross-section and branching fraction of b-hadron inclusive decays to the η c meson. A dedicated selection of η c mesons produced in b-hadron decays is developed to perform the most precise measurement of the ∆M J/ψ ,ηc . Both J/ψ and η c mesons are reconstructed via their decays to pp.
using a combination of many exclusive final states based on measurements from the B factories, Tevatron and LHC experiments [45].

Analysis technique
In this analysis, η c production is studied in a fiducial region of 6.5 < p T < 14.0 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/ψ meson [6,36]. Both the η c and the J/ψ mesons are reconstructed in the pp final state. The measured ratio is determined as σ where σ prompt ηc is the η c prompt production cross-section and σ b ηc is the production crosssection in inclusive b-hadron decays; N prompt ηc and N b ηc are the signal yields of η c mesons produced promptly and in b-hadron decays, respectively. Similar definitions apply for the J/ψ yields and cross-sections. The J/ψ ηc is the ratio of total efficiencies to trigger, reconstruct and select J/ψ → pp and η c → pp 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 η c and to J/ψ mesons, , is defined in the same way as for prompt production. The values of the branching fractions of the η c and J/ψ decays to pp, B ηc→pp = (1.50 ± 0.16) × 10 −3 and B J/ψ→pp = (2.120 ± 0.029) × 10 −3 , correspond to the current world-average values [36]. While the branching fraction of b-hadron inclusive decays to J/ψ meson, B b→J/ψ X = (1.16 ± 0.10) × 10 −3 , was measured at LEP [36], this analysis assumes the same value for the b-hadron mixture at LHC.
Since the masses of the η c and J/ψ states and kinematic distributions in J/ψ → pp and η c → pp decays are similar, they have similar reconstruction, trigger and selection efficiencies. Using simulation, the efficiency ratio of the J/ψ → pp and η c → pp modes is determined to be J/ψ / ηc = 1.00 ± 0.02, where the uncertainty is due to the size of the simulation samples. The efficiency ratio is also obtained in bins of p T , with negligible deviation from unity observed. Prompt J/ψ 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 decay time where ∆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 pp 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 fs, while a b-hadron-enriched sample is selected with t z > 80 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 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.

Fit to the invariant mass
A binned fit to the pp invariant mass of the prompt-enriched and b-hadron-enriched data samples is performed simultaneously in each bin of charmonium p T in order to extract the J/ψ signal yield and the η c -to-J/ψ yield ratio. For the η 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/ψ state is modelled by the sum of two Gaussian functions. In the study of the η c production, the mass values M J/ψ and ∆M J/ψ , ηc are constrained within uncertainties in each p 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 [36]. 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/ψ and η 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 η c model. A small p T -dependence of resolution parameters is seen in the simulation and is accounted for in the fit. The natural width of the η c meson is fixed to its known value [36]. 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 pp invariant mass spectrum. The only contribution relevant for this analysis is that from J/ψ → ppπ 0 decays, where the π 0 meson is not reconstructed. This background produces a broad, non peaking, contribution to the pp invariant mass below the known η c mass. In this region, the ppπ 0 background is described by a square-root shape, M T − M pp , below the phase-space limit, M T . The yield of this contribution is related to that of the decay J/ψ → pp by means of the ratio of branching fractions B J/ψ→ppπ 0 /B J/ψ→pp = 0.56±0.04 [36] and the ratio of efficiencies J/ψ→ppπ 0 / J/ψ→pp = 0.062 ± 0.002 for considered pp invariant mass window.
The pp invariant mass of selected candidates is shown in Fig. 1. Projections of the simultaneous fit result integrated over the entire p T range are overlaid. In general, the fit provides a good description of all M pp distributions. The charmonium yields in bins of p T and for the entire data sample are summarised in Table 1. These yields are corrected to take into account the cross-feed probabilities.

Systematic uncertainties
The systematic uncertainties on the η 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 T bins as a difference between the baseline fit result and the alternative fit result. Generally, bin-to-bin variations of uncorrelated systematic  The systematic uncertainties on the ⌘ c production corresponding to the signal and back-158 ground descriptions in the invariant-mass fit are estimated using alternative fit models.

159
Each uncertainty is estimated in p T bins as a di↵erence between the baseline fit result simulation. This systematic e↵ect is relevant for the di↵erential cross-section measurement.

166
The uncertainty corresponding to the combinatorial background description is estimated 167 by using an alternative model using a third-order polynomial function. The uncertainty as- uncertainties addressed in this paragraph are small compared to the statistical uncertainties. These variations are parametrised using a linear p T dependence to reduce statistical fluctuations. The uncertainty related to a potential p 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/ψ → ppπ 0 background contribution is estimated by varying the efficiency ratio J/ψ→ppπ 0 / J/ψ→pp and the branching fraction ratio B J/ψ→pp /B J/ψ→ppπ 0 [36] 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 Table 1: Yield of J/ψ mesons and the η c -to-J/ψ yield ratio for prompt and b-hadron decay production, corrected for the cross-feed, in bins of transverse momentum.  related to the ratio of η c and J/ψ 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 [49] with symmetric tails on both sides of the peak. The uncertainty on the η c natural width is accounted for by the difference in relative yields when using the world average value of 31.9 ± 0.7 MeV [36] and the value, 34.0 ± 1.9 ± 1.3 MeV, recently measured by LHCb collaboration [35]. Since this uncertainty is correlated among p T bins, the relative systematic uncertainty obtained from the p T -integrated data sample is taken as an estimate of the relative systematic uncertainty in each bin. This uncertainty is also correlated between p T bins. Possible nonzero polarisation of prompt J/ψ mesons affects their reconstruction efficiency. The J/ψ polarisation has not been measured at √ s =13 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/ψ simulation sample assuming polarisation parameter values λ θ = ±0.1 [50] in the pp collision frame. This uncertainty is correlated among p T bins. Systematic uncertainties on the relative cross-sections of the η 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 η 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 η c natural width and the invariant-mass resolution model. The uncertainty on the knowledge of the η 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.
Uncertainties on the branching fractions of the J/ψ → pp and η c → pp decays are considered separately. They are correlated among p T bins and amount to about 10%.   When deriving the absolute η c production cross-section, the uncertainty on the J/ψ production cross-section [6] is also taken into account.
6 Results and discussion on the η c η c η 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 Here and hereinafter, the quoted uncertainties are statistical, systematic and systematic due to uncertainties on the branching fractions B J/ψ→pp and B ηc→pp , respectively. Using the corresponding cross-section value of 0.749 ± 0.005 ± 0.028 ± 0.037 µb for prompt J/ψ production [6], the prompt η c production cross-section in the chosen fiducial region is derived to be σ prompt ηc 6.5<p T <14.0 GeV, 2.0<y<4.5 13 TeV = 1.26 ± 0.11 ± 0.08 ± 0.14 µb, where the last uncertainty includes in addition the uncertainty on the J/ψ production crosssection measurement. This is the first measurement of prompt η c production cross-section in proton-proton collisions at √ s =13 TeV, and it supports the conclusions of Ref.
[18] that suggests an enhanced η c production compared to that of the J/ψ meson. This measurement is in good agreement with the colour-singlet model prediction of 1.56 +0.83 Using the LHCb measurements of prompt η c production at the centre-of-mass energies √ s = 7 and 8 TeV [18], the prompt η c production cross-section dependence on the LHC energy is shown in Fig. 2. The J/ψ 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. The relative η c inclusive branching fraction from b-hadron decays is measured to be which combined with B b→J/ψ X = 1.16 ± 0.10% [36] gives B b→ηcX = (5.51 ± 0.32 ± 0.29 ± 0.77) × 10 −3 .
The last uncertainty includes the uncertainty on B b→J/ψ X . This result is the most precise measurement of the inclusive b → η 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 B ηc→pp and B b→J/ψ X . Numerical results of the measurements of p T -differential η c production are given in Appendix B. The relative η c to J/ψ p T -differential cross-sections for prompt and b-hadron decay production are compatible to those measured at √ s = 7 and 8 TeV [18] and are shown in Fig. 3. This is the first p T -differential cross-section measurement of η c prompt production at √ s = 13 TeV. The p T dependence of the prompt cross-section ratio is found to be linear with a slope of 0.22 ± 0.11 GeV −1 . While the integrated cross-section is in good agreement with the colour-singlet model prediction [31], a hint of a difference between the J/ψ and η c slopes motivates the extension of the measurement to larger p T values. A larger measured slope with respect to the prediction from Ref. [31] would indicate a possible colour-octet contribution. The absolute η c and J/ψ differential production cross-sections are shown in Fig. 4. The exponential slopes for the η c and J/ψ prompt differential cross-sections are determined from the fit to data points to be 0.41±0.07 GeV −1 and 0.57 ± 0.01 GeV −1 , respectively. /dp LHCb p s = 13 TeV 2.0 < y < 4.5 Figure 3: Relative ⌘ c to J/ di↵erential production cross-sections for (left) prompt production and (right) production in b-hadron inclusive decays. The uncertainties are statistical, systematic, and due to the ⌘ c ! pp and J/ ! pp branching fractions, respectively. For the relative prompt production cross-section, the result of a fit with a linear function is overlaid.  Figure 3: Relative η c to J/ψ 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 η c → pp and J/ψ → pp branching fractions, respectively. For the relative prompt production cross-section, the result of a fit with a linear function is overlaid. /dp LHCb p s = 13 TeV 2.0 < y < 4.5 Figure 3: Relative ⌘ c to J/ di↵erential production cross-sections for (left) prompt production and (right) production in b-hadron inclusive decays. The uncertainties are statistical, systematic, and due to the ⌘ c ! pp and J/ ! pp branching fractions, respectively. For the relative prompt production cross-section, the result of a fit with a linear function is overlaid. 7 Measurement of the J/ψ J/ψ J/ψ -η c η c η c mass difference While the prompt η 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 η c mass relative to the well-known J/ψ 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 GeV. The proton-antiproton system is required to have a vertex with a good fit quality, a large significance, χ 2 FD > 81, of the distance between this vertex and any     PV, and to have a transverse momentum greater than 5.5 GeV. The contamination of the selected sample from J/ψ and η c prompt production is estimated to be below 0.1%. The mass difference ∆M J/ψ , ηc is extracted from an extended maximum-likelihood fit to the M pp distribution. The fit provides a good description of the pp invariant-mass distribution (Fig. 5) yielding where the uncertainties are statistical and systematic.
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 T dependence.
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 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 [36], the recent BES III result [51], the latest BaBar measurement [52] and LHCb measurements [18, 32, 35].

Summary
Using data corresponding to an integrated luminosity of 2.0 fb −1 , the prompt η c production cross-section at a centre-of-mass energy of √ s = 13 TeV is measured for the first time.
The ratio of the prompt production rates of the η c and J/ψ states in the fiducial region where the quoted uncertainties are, in order, statistical, systematic and systematic due to uncertainties on the branching fractions, B J/ψ→pp and B ηc→pp . Using the prompt J/ψ production cross-section measurement at √ s = 13 TeV [6], the prompt η c production cross-section in the chosen fiducial region is derived to be σ prompt ηc 6.5<p T <14.0 GeV, 2.0<y<4.5 13 TeV = 1.26 ± 0.11 ± 0.08 ± 0.14 µb, where the last uncertainty includes in addition the uncertainty of the J/ψ production cross-section measurement. The result is in good agreement with the colour-singlet model prediction [31]. Contrary to NRQCD expectations, a steeper p T dependence of the J/ψ cross-section compared to that of the η c is preferred. The relative η c inclusive branching fraction from b-hadron decays is measured to be B b→ηcX /B b→J/ψ X = 0.48 ± 0.03 ± 0.03 ± 0.05.
Using B b→J/ψ X [36] the absolute η c inclusive branching fraction is obtained to be where the last uncertainty includes in addition the uncertainty on B b→J/ψ X . This result is consistent with the previous LHCb measurement [18]. Compatible results are obtained with an alternative analysis technique. The J/ψ -η c mass difference is measured using an enlarged data sample of b → η c X decays. The result, ∆M J/ψ , ηc = 113.0 ± 0.7 ± 0.1 MeV, is consistent with the world average value. It is the most precise η c mass determination to date.

Appendices A Alternative analysis based on t z t z 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 T and t z are obtained from a simultaneous fit to the corresponding pp invariant mass distributions of candidates in the bin. Prompt and b-decay charmonium contributions are then determined using a simultaneous χ 2 fit to the resulting t z distributions in p T bins, similarly to Ref. [6]. Projections of the simultaneous fit for the entire p T -range are shown in Fig. 6 for illustration purposes. In the alternative analysis discussed in this appendix, the pp invariant-mass fit is performed simultaneously in 28 two-dimensional bins of p 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 T bin. Promptly produced charmonium is distinguished from that produced in b-hadron decays by performing a simultaneous χ 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, τ b , is allowed to vary over p 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 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 T range are shown in Fig. 7.
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 η 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 T bins between factors of 0.5 and 1.5 of the estimated uncorrelated uncertainty. [ps] Candidates / ps   /dp LHCb p s = 13 TeV 2.0 < y < 4.5 Figure 8: Relative ⌘ c to J/ di↵erential 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 ⌘ c ! pp and J/ ! pp branching fractions. [ps] Candidates / ps   /dp LHCb p s = 13 TeV 2.0 < y < 4.5 Figure 8: Relative ⌘ c to J/ di↵erential 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 ⌘ c ! pp and J/ ! pp branching fractions. Figure 8: Relative η c to J/ψ 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 η c → pp and J/ψ → pp branching fractions.
B Tables of p T p T p T -differential η c η c η c production crosssections B.1 Prompt production of η c η c η c mesons The results on relative p T -differential η 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 T bins, and the last one is related to the B J/ψ→pp and B ηc→pp branching fractions. The results on p T -differential η c prompt production cross-section are shown in Table 6. Here, the last uncertainty includes the uncertainty on the J/ψ production cross-section. Table 5: Relative p T -differential η c prompt production cross-section.  Table 6: Differential η c prompt production cross-section. The results on relative p T -differential η c production in inclusive b-hadron decays are shown in Table 7. The results on p T -differential η 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/ψ production cross-section. Table 7: Relative p T -differential η c production cross-section in inclusive b-hadron decays.  Table 8: The p T -differential η c production cross-section in inclusive b-hadron decays.