Central exclusive production of $J/\psi$ and $\psi(2S)$ mesons in $pp$ collisions at $\sqrt{s}=13~$TeV

Measurements are reported of the central exclusive production of \jpsi and \psitwos mesons in $pp$ collisions at a centre-of-mass energy of 13 TeV. Backgrounds are significantly reduced compared to previous measurements made at lower energies through the use of new forward shower counters. The products of the cross-sections and the branching fractions for the decays to dimuons, where both muons are within the pseudorapidity range $2.0<\eta<4.5$, are measured to be $$ \begin{array}{rcl} \sigma_{J/\psi\rightarrow\mu^+\mu^-}&=&435 \pm 18 \pm 17 \pm 16 {\rm \ pb},\\ \sigma_{\psi(2S)\rightarrow\mu^+\mu^-}&=&11.1 \pm 1.1 \pm 0.3 \pm 0.4 {\rm \ pb}.\\ \end{array} $$ The first uncertainties are statistical, the second are systematic, and the third are due to the luminosity determination. The cross-sections are also measured differentially for meson rapidities between 2.0 and 4.5. Good agreement is observed with theoretical predictions. Photoproduction cross-sections are derived and compared to previous experiments, and a deviation from a pure power-law extrapolation of lower energy data is observed.


Introduction
Central exclusive production (CEP) [1] of a vector meson in pp collisions is a diffractive process in which the protons remain intact and the meson is produced through the fusion of a photon and a colourless strongly coupled object, the so-called pomeron. For charmonia production, the cross-section can be predicted in perturbative quantum chromodynamics (QCD) and at the leading order (LO) is proportional to the square of the gluon parton distribution function (PDF), which ensures a steep rise in the photoproduction cross-section with the centre-of-mass energy of the photon-proton system, W . Therefore, measurements of CEP of the J/ψ and ψ(2S) mesons provide not only a test of perturbative QCD but also probe the pomeron, and constrain the gluon PDF.
Elastic photoproduction of charmonia has been measured in fixed target experiments [2][3][4], in electron-proton [5][6][7][8], pp [9], and proton-lead collisions [10]. The LHCb collaboration has previously measured the CEP of the J/ψ and ψ(2S) mesons in pp collisions at a centre-of-mass energy √ s = 7 TeV [11]. In this paper, those results are extended to √ s = 13 TeV and charmonia are measured up to W = 2 TeV, the highest energy yet explored. This corresponds to probing the gluon PDF down to a fractional momentum of the proton, described by the Bjorken variable x ≈ 2 × 10 −6 , a scale at which saturation effects may become visible [12].
For diffractive processes, the dependence of the cross-section on the four-momentum transfer squared, t, is exponential with a slope b related to the transverse size of the interaction region. In Regge theory [13,14], b varies with W according to b = b 0 + 4α log(W/W 0 ), where b 0 is the slope measured at an energy W 0 . Measurements at HERA determined α = 0.164±0.041 GeV −2 with b 0 = 4.63 +0.07 −0.17 GeV −2 for J/ψ photoproduction at W 0 = 90 GeV [6]. In pp collisions at √ s = 7 TeV, LHCb measured b = 5.70 ± 0.11 GeV −2 at an average value of W = 750 GeV [11]. According to Regge theory, a value of b ≈ 6.1 GeV −2 is expected for J/ψ production in pp collisions at √ s = 13 TeV. In inelastic J/ψ production when proton dissociation occurs, the fall-off with t is more gradual. In contrast, the nonresonant ultraperipheral electromagnetic CEP of dimuons, produced through photon-photon fusion, peaks strongly at low t values. Therefore, the t dependence of the cross-section can be used to distinguish and study different production mechanisms.
This paper presents measurements of the cross-section for central exclusive production of charmonia with rapidity, y, between 2.0 and 4.5, and follows the methodology of the LHCb analysis at √ s = 7 TeV [11]. Exclusive charmonium candidates are selected through their characteristic signature at a hadron collider: a pp interaction devoid of any activity save the charmonium that is reconstructed from its decay to two muons. The addition of new forward shower counters (HeRSCheL) [15] extends the pseudorapidity region in which particles can be vetoed and roughly halves the number of background events compared to the previous measurement.
The LHCb detector is outlined in Sec. 2 while the data and selection criteria are described in Sec. 3. The cross-section calculation is detailed in Sec. 4 and systematic uncertainties are presented in Sec. 5. The cross-section results for pp → pJ/ψ p and pp → pψ(2S)p processes and derived photoproduction cross-sections for γp → J/ψ p and γp → ψ(2S)p are presented in Sec. 6. Conclusions are given in Sec. 7.

Detector, data samples and triggers
The LHCb detector [16,17] is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 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 (VELO) surrounding the pp interaction region, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about 4 Tm, and three stations of silicon-strip detectors and straw drift tubes placed downstream of the magnet. The tracking system provides a measurement of momentum, p, of charged particles with a relative uncertainty that varies from 0.5% at low momentum to 1.0% at 200 GeV. 1 Photons, electrons and hadrons are identified by a calorimeter system consisting of scintillating-pad (SPD) and preshower detectors, an electromagnetic calorimeter and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers [18].
The pseudorapiditry coverage is extended by forward shower counters consisting of five planes of scintillators with three planes at 114, 19.7 and 7.5 m upstream of the interaction point, and two downstream at 20 and 114 m. At each location there are four quadrants of scintillators, whose information is recorded in every beam crossing by photomultiplier tubes, giving a total of 20 channels in HeRSCheL. These are calibrated using data taken without beams circulating at the end of each LHC fill. The pseudorapidity ranges covered by VELO and HeRSCheL are different. For VELO, the region is −3.5 < η < −1.5 and 2 < η < 5, and for HeRSCheL, the region is −10 < η < −5 and 5 < η < 10.
A data set corresponding to an integrated luminosity of 204 ± 8 pb −1 in pp collisions at √ s = 13 TeV is used in this analysis. The average number of pp interactions per beam crossing, µ, is 1.1, thus in about half of visible interactions there is only a single pp collision and the CEP process is uncontaminated by pile-up. The online event selection is performed by a trigger that consists of two different stages. First, there is a hardware stage, which requires less than 30 deposits in the SPD and at least one muon with a transverse momentum, p T , above 200 MeV. It is followed by a software stage, which applies a full event reconstruction and requires fewer than ten reconstructed tracks, at least one of which is identified as a muon.
Simulated signal events are generated using SuperCHIC v2.02 [19], where the J/ψ and ψ(2S) mesons are transversely polarised. The J/ψ meson can also originate from exclusive χ c decays, which are also generated with SuperCHIC, or from ψ(2S) decays, which are handled by PYTHIA [20]. The LPAIR generator [21] is used to generate dimuons produced through the electromagnetic photon-photon fusion process. The interaction of the generated particles with the detector, and the detector response, are implemented using the Geant4 toolkit [22] as described in Ref. [23].

Event selection
The selection of candidate signal events is similar to that used in the previous LHCb analysis [11]. Two reconstructed muons are required in the region 2.0 < η < 4.5, with an invariant mass within ±65 MeV of the known J/ψ or ψ(2S) mass [24] and p 2 T of the reconstructed meson below 0.8 GeV 2 . Events with additional VELO tracks or photons with transverse energies above 200 MeV are vetoed. Events with significant deposits in HeRSCheL are removed. The HeRSCheL response is described using a variable χ 2 HRC that quantifies the activity above noise, taking account of correlations between the counters.
The invariant mass of all candidates without the mass-window requirement applied is shown in Fig. 1. The data in the nonresonance regions are candidates for electromagnetic CEP dimuons produced by photon-photon fusion and constitute an important calibration sample. The p 2 T distribution of these dimuons with and without the requirement on χ 2 HRC is shown in Fig. 2 and is significantly peaked towards low values due to the long-range electromagnetic interaction. The fraction of electromagnetic CEP events in this sample is determined from a fit to the p 2 T distribution with two components: a signal shape taken from simulated events and an inelastic background modelled with the sum of two exponential functions.
The power of HeRSCheL to discriminate CEP events can be seen in Fig. 3, which shows the distributions of χ 2 HRC for three classes of low-multiplicity-triggered events. The first class is CEP-enriched dimuons: events in the nonresonant dimuon sample with p 2 T < 0.01 GeV 2 , which has a purity of 97% for electromagnetic CEP events. The second class, inelastic-enriched J/ψ , applies the nominal J/ψ selections but requires p 2 T > 1 GeV 2 , thus selecting inelastic events with proton dissociation. The third class consists of events with more than four tracks reconstructed. Figure 3 shows that CEP-enriched events have lower values of χ 2 HRC . To select exclusive J/ψ and ψ(2S) candidates, it is required that log(χ 2 HRC ) < 3.5; this value is chosen in order to minimise the combined statistical and systematic uncertainty on the total cross-sections. After the event selections, there are 14 753 J/ψ signal candidates and 440 ψ(2S) signal candidates remaining.
The estimation of the signal efficiency, H , for the requirement log(χ 2 HRC ) < 3.5 is described in Sec. 3.1. Using this, Sec. 3.2 explains how the purity of the signal sample is estimated. The signal efficiency of all selection requirements is detailed in Sec. 3.3.   HRC that is related to activity in HeRSCheL. The response to three classes of events, as described in the text, is shown. The selection requirement for the analysis is indicated by the red vertical line and the arrow.

HeRSCheL efficiency of selecting signal events
The efficiency for the veto on HeRSCheL activity is estimated from data using the nonresonant calibration sample. The fits to the p 2 T distributions in Fig. 2 give the numbers of electromagnetic CEP events with and without the HeRSCheL veto. The ratio of these gives the efficiency of the veto, which is determined to be H = 0.723 ± 0.008. The signal loss includes in particular a contribution from events where there is an additional primary interaction only seen in the HeRSCheL detector, as well as spill-over from previous collisions, electronic noise and calibration effects, as discussed in Ref. [15]. This efficiency, measured using the nonresonant sample, is applicable to any CEP process, with the same veto, collected in this data-taking period.

Purity of signal sample
Three background sources are considered: nonresonant dimuon production; feed-down of CEP χ cJ (1P ) or ψ(2S) to J/ψ mesons and other undetected particles; and nonexclusive events where the proton dissociates but the remnants remain undetected.
The amount of nonresonant background is determined from the fit shown in Fig. 1, where the signals are modelled with two Crystal Ball functions [25] and the nonresonant background with the sum of two exponential functions. This background is estimated to contribute a fraction of 0.009 ± 0.001 to the J/ψ and 0.161 ± 0.018 to the ψ(2S) samples.
The ψ(2S) feed-down background in the J/ψ selection is determined using simulated events that have been normalised to have the same yield as the ψ(2S) → µ + µ − signal in data and is estimated to contribute a fraction 0.015 ± 0.001 to the J/ψ samples. The χ cJ (1P ) feed-down background is determined using a data calibration sample, which contains events that pass the nominal J/ψ selection, except instead of zero photons, it is required that there is exactly one reconstructed photon with a transverse energy above 200 MeV. The numbers of χ c0 (1P ), χ c1 (1P ), and χ c2 (1P ) candidates in this calibration sample are determined from a fit to the invariant mass of the dimuon plus photon system. These are scaled by the ratio of J/ψ to J/ψ + γ candidates in the corresponding simulated χ cJ (1P ) sample from which it is esimated that a fraction of 0.005 ± 0.001 of the J/ψ candidate sample is due to feed-down from χ c0 (1P ) mesons, 0.002 ± 0.001 from χ c1 (1P ) mesons, and 0.038 ± 0.002 from χ c2 (1P ) mesons. The total feed-down ratio from ψ(2S) and χ cJ (1P ) mesons is 0.060 ± 0.002, to be compared to 0.101 ± 0.009 in the previous analysis [11]: the addition of HeRSCheL suppresses events with proton dissociation, which are more numerous in the double-pomeron-exchange process that mediates χ cJ (1P ) production.
The fraction of nonexclusive events due to proton dissociation is determined through the p 2 T distribution of the J/ψ and the ψ(2S) candidates, after a background subtraction to remove contributions coming from the electromagnetic nonresonant and feed-down backgrounds. The electromagnetic component is shown in Fig. 2, while the feed-down shape is taken from the J/ψ + γ calibration sample. The background-subtracted p 2 T distribution consists of two remaining components: signal and proton dissociation background. Since t ≈ −p 2 T , approximately exponential distributions with different slopes are expected for each. In the previous analysis [11], each was modelled by an exponential function whose slope was a free parameter. The presence of the HeRSCheL detector however now allows these shapes to be determined from data, thus reducing the model dependence of the result.
The background subtracted distribution without HeRSCheL veto applied is split into two distributions: S H if log(χ 2 HRC ) < 3.5 (corresponding to the signal selection), and SH otherwise. Since H and (1 − H ) are the respective efficiencies for a CEP event to enter the distributions S H and SH, the distribution, β = SH − ((1 − H )/ H )S H , by construction has no contribution coming from exclusive events. The distribution for β approximates to the shape of the proton dissociation in the candidate distribution S H , but is not exactly the same since the efficiency to veto nonexclusive events has a weak dependence on p 2 T .  Consequently, the proton dissociation in the distribution S H is estimated by scaling the distribution β by f (p 2 T ) ≡ S H (p 2 T )/β(p 2 T ). The scale factor f (p 2 T ) is known from data for values of p 2 T 0.8 GeV 2 , since there is little signal in this region as the signal distribution is expected to follow exp(−b sig p 2 T ) with b sig ≈ 6 GeV −2 . An extrapolation of f (p 2 T ) is performed to the region p 2 T < 0.8 GeV 2 using functions which fit the data well in the region p 2 T > 0.8 GeV 2 . The default is an exponential function for the J/ψ analysis and a constant for the ψ(2S) analysis. A linear dependence is used to estimate the systematic uncertainty.
The p 2 T candidate distributions in data with the estimated backgrounds superimposed are shown in the upper row of Fig. 4. The lower row shows the signal components after subtracting the proton dissociation background. These are fitted with a single exponential function, exp(−b sig p 2 T ), to test the hypothesis that the signal has this dependence. The J/ψ signal contribution is well described with b sig = 5.93 ± 0.08 GeV −2 , consistent with extrapolations from previous pp measurements at 7 TeV and from H1 results [5,11]. The corresponding slope, in the ψ(2S) analysis, is b sig = 5.06 ± 0.45 GeV −2 . Fits to the derived proton dissociation components show that these are also consistent with a single exponential.
In the region 0 < p 2 T < 0.8 GeV 2 , 0.175±0.015 of the J/ψ candidate sample is estimated to be due to proton-dissociation events, while for the ψ(2S) sample the contamination is estimated to be 0.11 ± 0.06. The uncertainties are statistical, and the correlation between the HeRSCheL efficiency and the proton-dissociation contamination is taken into account. The current analysis shows an approximate halving of the proton-dissociation background compared to the analysis at √ s = 7 TeV, due to the additional HeRSCheL veto. The overall purities are 0.755 ± 0.015 and 0.726 ± 0.061 for the J/ψ and ψ(2S) selections, respectively.

Selection efficiency
The efficiency for selecting signal events is the product of the reconstruction efficiency, rec , and selection efficiency, sel . The reconstruction efficiency is the product of trigger, tracking, and muon identification efficiencies. Each is determined from simulation and scaled using a data calibration sample. The trigger efficiency is calibrated through the fraction of events where both muons pass the trigger, in a sample collected with the requirement that at least one muon passes the trigger. The muon identification efficiency is calibrated using a sample enriched in J/ψ mesons that has been selected requiring a single identified muon. The tracking efficiency is calibrated using low-multiplicity events where the dimuon hardware was triggered by two objects having an absolute azimuthal angular difference close to π.
The efficiency for the selection requirements on the mass and transverse momentum of the J/ψ candidate, and the veto on additional tracks, photon activity, or HeRSCheL activity is obtained from data.
The fits to the mass distributions in Fig. 1 determine the fraction of signal inside the mass window and give a signal efficiency of 0.967 ± 0.002. No dependence on rapidity is found.
The efficiency for the requirement on the meson candidates that p 2 T < 0.8 GeV 2 is 0.993 ± 0.001 and is determined from the fitted slope to the signal components shown in Fig. 4 as described in the previous section. A small dependence on rapidity y is introduced through the Regge extrapolation of the exponential slope: s. The signal efficiency of vetoing events with additional VELO tracks or photons is obtained using the same technique described in Sec. 3.1 to determine the HeRSCheL veto efficiency. When vetoing events with additional VELO tracks, no dependence on rapidity is found in simulation, while a slight dependence is observed for the photon veto, which is due to material effects in the detector whose density varies with rapidity. The shape of the rapidity dependance is taken from simulation and normalised to data. The efficiency of vetoing events with VELO tracks is determined to be 0.969 ± 0.004 and of vetoing events with photons is on average 0.983 ± 0.003.

Cross-section calculation
The products of the cross-sections and the branching fractions of the decays to two muons, σ ψ→µµ , are measured differentially in ten equally spaced bins of J/ψ rapidity and three unequal bins of ψ(2S) rapidity in the range y ∈ (2.0, 4.5). The measurements are limited to the fiducial region where both muons have pseudorapidities between 2.0 and 4.5.
The differential cross-section in each bin is and the total cross-section, summed over all bins, is also calculated. In Eq. 1, N is the number of selected events, rec and sel are the efficiencies described in Sec. 3.3, P is the purity given in Sec. 3.2 and ∆y is the width of the rapidity bin. The effective luminosity is the product of the integrated luminosity, L tot , and the fraction of single interaction beam crossings, single . The number of visible pp interactions, v, is assumed to follow a Poisson distribution, P (v) = µ v e −µ /v!. The mean µ is determined from the fraction of beam crossings with no visible activity, and single = µe −µ , which is calculated over the datataking period in roughly hour-long intervals. The average value is single = 0.3662 ± 0.0003, which is about 50% higher than the corresponding value in the 7 TeV analysis. The lower number of pp interactions per beam crossing at √ s = 13 TeV benefits the collection of CEP events. The integrated luminosity, L tot , is evaluated as 204 ± 8 pb −1 and is found from µ and a constant of proportionality that is measured in a dedicated calibration dataset [26].

Systematic uncertainties
Various sources of systematic uncertainties have been considered and are summarised in Table 1 for the total cross-section. Excluding the uncertainty on the luminosity, they amount to 2.5% in the J/ψ and 2.7% in the ψ(2S) cases.
The largest source of systematic uncertainty comes from the determination of the HeRSCheL efficiency. The fit to the p 2 T distribution in Fig. 2 depends on assumptions made on the shape of the signal and background components. A systematic uncertainty is assessed firstly by changing the functional form of the background description, secondly by fitting only the tail of the distribution and extrapolating the result to the signal, and thirdly by using only the candidates in the first bin of the p 2 T distribution where the signal dominates. The differences of each to the nominal fit are combined in quadrature which results in a systematic uncertainty of 1.7% on the total cross-section.
Since the same methodology is used to determine the efficiency for vetoing events with additional VELO tracks or photons, the associated systematic uncertainty is estimated with the same procedure. Since the simulation shows a dependence on rapidity for the efficiency due to the photon requirement, an additional uncertainty is added in quadrature in each rapidity bin, corresponding to the limited sample size of the simulation. This leads to a total systematic uncertainty of 0.2% on the total cross-section due to each veto requirement.
The systematic uncertainty on the efficiency of the mass-window requirement is obtained by repeating the fit shown in Fig. 1 with the mass peak and resolution fixed to the values of the simulation. The fit is also repeated by changing the background description to a single exponential function across the whole region. The biggest difference with the nominal fit between these two alternative fits is taken as the systematic uncertainty, which is 0.6% on the total cross-sections.
The uncertainty on the efficiency of selecting candidates with p 2 T < 0.8 GeV 2 is 0.3%. It is obtained by varying the signal shape from that shown in Fig. 4 to the one obtained by using the approach of the previous analysis [11] where the p 2 T distribution is fitted with two exponential functions, one describing the proton dissociation and the other the signal shape. The slope and normalisations of each are free. The difference in efficiency between the two approaches is added in quadrature to the uncertainty coming from the propagation of the uncertainties on the parameters describing the Regge dependence that determines the rapidity dependence.
The proton-dissociation contamination depends on the extrapolation from the background-dominated high p 2 T region to the signal-dominated low p 2 T region. The corresponding systematic uncertainty is assigned by changing the form of the extrapolation function from the nominal exponential one to an alternative linear function, or fitting the p 2 T distribution with two exponential functions to get the background contamination. The systematic uncertainty is the biggest difference between the nominal results and those from the two alternative approaches, and corresponds to 0.7% on the total cross-section.
The systematic uncertainty due to the feed-down contribution in the J/ψ analysis is assessed to be 0.7% on the total cross-section. It corresponds to the largest difference in the cross-section determination from a series of alternative fits to the J/ψ + γ spectrum in which the photon energy scale, photon detection efficiency, invariant mass resolution, material interactions, and the ψ(2S) contribution, are each varied by their estimated uncertainties.
An alternative estimate of the nonresonant background in Fig. 1 is performed by fitting a single exponential function between 1.5 and 2.5 GeV and extrapolating this into the signal region. This changes the total cross-section by 0.1% in the J/ψ analysis and 1.5% in the ψ(2S) analysis. These values are taken as systematic uncertainties due to the nonresonant background.
The reconstruction efficiency is taken from simulated events and calibrated using data. The technique depends on tagging a muon that fired the trigger and probing a partially reconstructed track that forms a J/ψ candidate. To assess the systematic uncertainty due to the method, this technique is applied to two simulated samples that have different tracking efficiencies. The resulting tracking efficiencies are compared after calibration using data. In a second test of the methodology, one simulated sample is taken as pseudodata and the other simulated sample applies the calibration procedure. The resulting efficiencies are compared to the true values in the pseudodata. The largest difference in each rapidity bin is assigned as a systematic uncertainty, which is assumed to be fully correlated between bins, and varies from 0.5% to 3.1% depending on the sample size. A systematic uncertainty on the method used in evaluating the muon identification and trigger efficiencies is assigned by comparing the derived values in simulation with truth, resulting in a 0.4% uncertainty on the total cross-section due to the muon identification, and 0.2% due to the trigger.
A bin migration uncertainty has been estimated using simulation to relate the reconstructed and true rapidity bin. The difference is smaller than 0.06% in all bins and so is considered negligible.
Most systematic uncertainties are assumed to be 100% correlated between rapidity bins except the photon-veto-shape systematic uncertainty, which is assumed to be independent between bins as it depends on the statistical precision of the simulation. As the determination of the sample purity depends on the HeRSCheL efficiency, these two quantitities are correlated. The correlation factors are determined in simulation, and the values are ρ = −0.50 and ρ = −0.06 for the J/ψ and ψ(2S) selection, respectively. The lower statistical precision of the ψ(2S) sample imposes less constraint on the proton dissociation scale factor f (p 2 T ) and results in a smaller correlation. The total systematic uncertainties are given in Table 1 taking account of the correlations.

Results
The product of the differential cross-sections and branching fractions to two muons, with both muons inside the fiducial acceptance 2.0 < η < 4.5, are given per meson rapidity bin in Tables 2 and 3 for J/ψ and ψ(2S) mesons, respectively. The tables also present a summary of the numbers entering the cross-section calculation. The correlations between the statistical and systematic uncertainties in each bin are shown in Tables 5 and 6 in the Appendix. Summing these differential results leads to measurements of the product of the cross-sections and branching fractions, where both muons are within the fiducial region, 2.0 < η < 4.5: The first uncertainties are statistical and include the uncertainties on the data-driven efficiencies and purities, the second are systematic, and the third are due to the luminosity determination.
As a cross-check and to confirm the improvements brought by HeRSCheL, the crosssections have been recalculated without imposing the HeRSCheL veto: consistent results are obtained but with a larger systematic uncertainty of about 8%. While the extracted signal contribution is comparable to Fig. 4 and well described by a single exponential function with a consistent value of b sig = 5.92 ± 0.06 GeV −2 , the extracted protondissociation component requires two exponential functions to describe the distribution.  3.9 3.9 3.9 Table 4: Tabulation, in bins of meson rapidity, of the fraction of decays with both muons in the range 2.0 < η < 4.5 and the differential cross-sections for J/ψ and ψ(2S) production calculated without fiducial requirements on the muons. To compare with theoretical predictions, which are generally expressed without fiducial requirements on the muons, the differential cross-sections for J/ψ and ψ(2S) mesons as functions of the meson rapidity are calculated by correcting for the branching fractions to muon pairs, B(J/ψ → µ + µ − ) = (5.961 ± 0.033)% and B(ψ(2S) → µ + µ − ) = (0.79 ± 0.09)% [24], and for the fraction of those muons that fall inside the fiducial acceptance of the measurement. The fiducial acceptance is determined using SuperCHIC [19] assuming that the polarisation of the meson is the same as that of the photon. The acceptance values in bins of meson rapidity are tabulated in Table 4 along with the differential cross-section results. These are plotted in Fig. 5 and compared to the theoretical calculations of Refs. [27,28]. Both measurements are in better agreement with the next-to-LO (NLO) predictions. The χ 2 /ndf for the J/ψ analysis is 8.5/10 while for the ψ(2S) analysis, it is 4.9/3. They are less consistent with the LO predictions having 32.8/10 and 15.4/3 for the J/ψ and ψ(2S) analysis, respectively.
The cross-section for the CEP of vector mesons in pp collisions is related to the photoproduction cross-section, σ γp→ψp [27], Here, r is the gap survival factor, k ± ≡ M ψ /2e ±y is the photon energy, dn/dk ± is the  [27,28] for the J/ψ meson (top) and the ψ(2S) meson (bottom). The inner error bar represents the statistical uncertainty; the outer is the total uncertainty. Since the systematic uncertainty for the ψ(2S) meson is negligible with respect to the statistical uncertainty, it is almost not visible in the lower figure.
photon flux and W 2 ± = 2k ± √ s is the invariant mass of the photon-proton system. Equation 2 shows that there is a two-fold ambiguity with W + , W − both contributing to one LHCb rapidity bin. Since the W − solution contributes about one third and as it has been previously measured at HERA, this term is fixed using the H1 parametrisation of their results [5]: σ γp→J/ψp = a(W/90 GeV) δ with a = 81 ± 3 pb and δ = 0.67 ± 0.03. For the ψ(2S) W − solution, the H1 J/ψ parametrisation is scaled by 0.166, their measured ratio of ψ(2S) to J/ψ cross-sections [8]. The photon flux is taken from Ref. [29] and the gap survival probabilities are taken from Ref. [30]. With these inputs, which for ease of calculation are reproduced in Tables 7 and 8 in the Appendix, Eq. 2 allows the calculation of σ γp→ψp at high values of W beyond the kinematic reach of HERA.
The photoproduction cross-sections for J/ψ and ψ(2S) are shown in Fig. 6. It includes a comparison to H1 [5], ZEUS [7] and ALICE [10] results, and at lower W values fixed target data from E401 [2], E516 [3] and E687 [4]. Also shown are previous LHCb results at √ s = 7 TeV, recalculated using improved photon flux and gap survival factors. The 13 TeV LHCb data are in agreement with the 7 TeV results in the kinematic region where they overlap. However, the 13 TeV data extends the W reach to almost 2 TeV. Figure 6 also shows the power-law fit to H1 data [5] and it can be seen that this is insufficient to describe the J/ψ data at the highest energies. In contrast, the data is in good agreement with the JMRT prediction, which takes account of most of the NLO QCD effects [30] and deviates from a simple power-law shape at high W .

Conclusions
Measurements are presented of the cross-sections times branching fractions for exclusive J/ψ and ψ(2S) mesons decaying to muons with pseudorapidities between 2.0 and 4.5. The addition of new scintillators in the forward region has resulted in lower backgrounds in pp collisions at a centre-of-mass energy √ s = 13 TeV compared to the previous measurement at √ s = 7 TeV. As a consequence, the systematic uncertainty is reduced from 5.6% at √ s = 7 TeV to 2.7% at √ s = 13 TeV, reflecting an improved understanding of the background proton-dissociation process. After correcting for the muon acceptance, the cross-sections for the J/ψ and ψ(2S) mesons are compared to theory and found to be in better agreement with the JMRT NLO rather than LO predictions. The derived crosssection for J/ψ photoproduction shows a deviation from a pure power-law extrapolation of H1 data, while the ψ(2S) results are consistent although more data are required in this channel to make a critical comparison.
LHCb collaboration 16 Table 5: (Top) Statistical and (bottom) systematic correlation matrices for J/ψ , where each column corresponds to one rapidity bin in increasing order. As the matrix is symmetric, only the top triangle is shown. Table 7: Values used in evaluating the photo-production cross-section using Eq. 2 for the J/ψ and ψ(2S) analysis with gap survival factors for the production of J/ψ and ψ(2S) mesons at √ s = 13 TeV [30]. For the J/ψ analyis, σ γp→J/ψp(W + ) is calculated using the power-law description of HERA or the JMRT NLO description for σ γp→J/ψp(W − ) .