Updated measurements of exclusive $J/\psi$ and $\psi(2S)$ production cross-sections in $pp$ collisions at $\sqrt{s}=7$ TeV

The differential cross-section as a function of rapidity has been measured for the exclusive production of $J/\psi$ and $\psi(2S)$ mesons in proton-proton collisions at $\sqrt{s}=7$ TeV, using data collected by the LHCb experiment, corresponding to an integrated luminosity of 930 pb$^{-1}$. The cross-sections times branching fractions to two muons having pseudorapidities between 2.0 and 4.5 are measured to be $$\begin{array}{rl} \sigma_{pp\rightarrow J/\psi\rightarrow{\mu^+}{\mu^-}}(2.0<\eta_{\mu^\pm }<4.5)=&291\pm 7\pm19 {\rm \ pb},\\ \sigma_{pp\rightarrow\psi(2S)\rightarrow{\mu^+}{\mu^-}}(2.0<\eta_{\mu^\pm}<4.5)=&6.5\pm 0.9\pm 0.4 {\rm \ pb},\end{array}$$ where the first uncertainty is statistical and the second is systematic. The measurements agree with next-to-leading order QCD predictions as well as with models that include saturation effects.


Introduction
Exclusive J/ψ and ψ(2S) meson production in hadron collisions are diffractive processes that can be calculated in perturbative quantum chromodynamics (QCD) [1]. At leading order they are thought to proceed via the exchange of a photon and a pomeron, which at sufficiently hard scales can be described by two gluons as shown in Fig. 1(a). Measurements of exclusive J/ψ and ψ(2S) production thus provide a test of QCD and shed light on the pomeron, which plays a critical role in the description of diffraction and soft processes. In particular, the measurements are sensitive to saturation effects [2,3]: when performed in the pseudorapidity range of the LHCb detector they probe x, the fractional momentum of the parton, down to 5 × 10 −6 . Since the theoretical predictions depend on the gluon parton density function (PDF), the experimental measurements can be used to constrain it [4,5]. Furthermore, the measurements are sensitive to the presence of the odderon [6], the odd-parity partner of the pomeron, which could mediate the reaction in place of the photon in Fig. 1(a).
Several measurements of exclusive J/ψ production have been reported by the H1 [7,8] and ZEUS [9] collaborations at the HERA ep collider, at values of W , the centre-of-mass energy of the photon-proton system, between 30 and 300 GeV. The first measurement at a hadron machine, at W ≈ 80 GeV, was made by the CDF collaboration [10] at the Tevatron pp collider. The first measurement in pp collisions was made by the LHCb collaboration [11] using data corresponding to an integrated luminosity of 37 pb −1 collected at √ s = 7 TeV, and this extended the W reach up to 1.5 TeV. Measurements in Pb-Pb collisions at the LHC have been reported by the ALICE collaboration [12]. Measurements of ψ(2S) production have been made by the H1 collaboration [13] at four W values while both CDF [10] and LHCb [11] reported results using small samples of ψ(2S) consisting of about 40 candidates each.
The J/ψ photoproduction cross-section has been fit by a power-law function, σ γp→J/ψ p (W ) = a(W/90 GeV) δ , with the H1 collaboration measuring a = 81 ± 3 pb and δ = 0.67 ± 0.03 [8]. At leading order this follows from the small-x parametrisation of the gluon PDF: g(x, Q 2 ) ∝ x λ at the scale Q 2 = M 2 J/ψ /4, where M J/ψ is the mass of the J/ψ meson. All measurements to date at hadron machines are consistent with this, albeit with rather large uncertainties. However, higher-order corrections [5] or saturation effects [2,3] lead to deviations from a pure power-law behaviour and the measurements presented here have sufficient precision to probe this effect. The ψ(2S) differential cross-section measurements from the H1 collaboration are also consistent with a power-law function, although the limited data sample implies a rather large uncertainty and leads to a value for the exponent of δ = 0.91 ± 0.17 [13]. Both CDF and LHCb results are consistent with this.
This paper presents updated measurements from the LHCb collaboration using 930 pb −1 of data collected in 2011 at √ s = 7 TeV. Both the J/ψ and ψ(2S) cross-sections are measured differentially as a function of meson rapidity and compared to various theoretical models, including those with saturation effects. The analysis technique is essentially that published previously [11]. The main difference concerns the methodology for determining the background due to non-exclusive J/ψ and ψ(2S) production where the additional particles remain undetected.

Detector and data samples
The LHCb detector [14] is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 5 (forward region), 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 (TT) located upstream of a dipole magnet with a bending power of about 4 Tm, and three stations of silicon-strip detectors (IT) and straw drift-tubes (OT) [15] placed downstream. The combined tracking system provides a momentum measurement with relative uncertainty that varies from 0.4% at 5 GeV/c to 0.6% at 100 GeV/c, and impact parameter resolution of 20 µm for tracks with large transverse momentum. Different types of charged hadrons are distinguished by information from two ring-imaging Cherenkov detectors [16]. Photon, electron and hadron candidates are identified by a calorimeter system consisting of scintillating-pad (SPD) and pre-shower detectors, an electromagnetic calorimeter and a hadronic calorimeter. The SPD also provides a measurement of the charged particle multiplicity in an event. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers [17]. The trigger [18] 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 VELO also has sensitivity to charged particles with momenta above ∼100 MeV/c in the pseudorapidity range −3.5 < η < −1.5 (backward region), while extending the sensitivity of the forward region to 1.5 < η < 5.
The J/ψ and ψ(2S) are identified through their decay to two muons. The protons are only marginally deflected by the peripheral collision and remain undetected inside the beam pipe. Therefore, the signature for exclusive vector meson production is an event containing two muons and no other activity. Beam-crossings with multiple proton interactions produce additional activity; in the 2011 data-taking period the average number of visible interactions per bunch crossing was 1.4. Consequently, requiring an exclusive signature restricts the analysis to beam crossings with a single pp interaction.
The SuperChic [19] generator is used to produce samples of exclusive J/ψ and ψ(2S) decays as well as those of the χ c meson, which form a background for the J/ψ analysis. These are passed through a GEANT4 [20] based detector simulation, the trigger emulation and the event reconstruction chain of the LHCb experiment.

Event selection
The hardware trigger used in this analysis requires a single muon track with transverse momentum p T > 400 MeV/c in coincidence with a low SPD multiplicity (< 10 hits). The software trigger used to select signal events requires two muons with p T > 400 MeV/c.
The analysis is performed in ten equally sized bins of meson rapidity between 2.0 and 4.5. The selection of exclusive events begins with the requirement of two reconstructed muons in the forward region. The acceptance of LHCb for muons from J/ψ and ψ(2S) decays is not uniform: muons with low momenta can be swept out of the LHCb acceptance by the magnetic field, or be absorbed before they reach the muon stations. Consequently, a fiducial region is defined requiring that each muon has a momentum greater than 6 GeV/c and both tracks are reconstructed within the muon chamber acceptance. The fiducial acceptance is determined using simulated events and is shown in Fig. 2 for both J/ψ and ψ(2S) decays, both of which are assumed to be transversely polarised due to s-channel helicity conservation. A systematic uncertainty of 2%, fully correlated between bins, is taken on these values corresponding to the estimated uncertainty on the description of the tracking in the simulation [21].
It is required that there are no photons reconstructed in the detector and no other tracks that incorporate VELO information. Unlike the previous analysis [11], a veto is not imposed on tracks constructed solely from TT and IT/OT detector combinations, as these  can arise from detector signals associated with previous beam interactions ("spillover"). The absence of activity apart from the two muons ensures two rapidity gaps which sum to 3.5 units in the forward region. An additional rapidity gap is obtained by requiring that there are no tracks in the backward region. The VELO is sensitive to tracks within a certain rapidity region depending on the z position from which the tracks originate and the event topology. The mean size of the backward rapidity gap that can be identified is 1.7 with a root mean square of about 0.5.
Muon pairs are combined to form meson candidates whose transverse momentum squared must satisfy p 2 T < 0.8 GeV 2 /c 2 , and whose invariant masses must lie within 65 MeV/c 2 of the known J/ψ or ψ(2S) mass values [22]. With these requirements, 55,985 J/ψ candidates and 1565 ψ(2S) candidates are found.
Three background components are considered: non-resonant background due primarily to the QED process that produces two muons; feed-down from exclusive production of other mesons (e.g. χ c ); and inelastic production of mesons where one or both protons disassociate.

Non-resonant background determination
The invariant mass distributions for J/ψ and ψ(2S) candidates are shown in Fig. 3 and are fitted with Crystal Ball functions [23] to describe the resonant contributions and an exponential function for the non-resonant background. Within the range of ±65 MeV/c 2 about the known meson masses, the non-resonant background is estimated to account for (0.8 ± 0.1)% and (17.0 ± 0.3)% of the J/ψ and ψ(2S) candidates, respectively.

Feed-down background determination
Exclusively produced χ c or ψ(2S) mesons can feed down to mimic an exclusive J/ψ decay when the particles produced in association with the J/ψ remain undetected or go outside the detector acceptance. Their contribution is estimated using simulated events normalised to an enriched background sample in the data. Exclusive χ c candidate events are identified in the data as those containing a J/ψ and a single photon [24]. The background from χ c feed-down is then estimated by scaling the number of observed χ c candidates by the ratio of simulated χ c mesons passing the J/ψ selection requirements compared to those identified as χ c candidate events. The feed-down from χ c decays is estimated to account for (7.6 ± 0.9)% of the exclusive J/ψ candidates, where the uncertainty includes a contribution from the fitted proportions of χ c0 , χ c1 , χ c2 as well as the photon reconstruction efficiency in simulation. Feed-down from ψ(2S) decays is estimated by scaling the ψ(2S) yields in the resonant peak ( Fig. 3) by the ratio of simulated ψ(2S) mesons passing the J/ψ selection requirements compared to those passing the ψ(2S) selection requirements. The feed-down from ψ(2S) decays is estimated to account for (2.5 ± 0.2)% of the exclusive J/ψ candidates.
Feed-down into the ψ(2S) selection is expected to be very small, e.g. due to χ c (2P ) or X(3872) decays [25,26]. Relaxing the requirement on the number of photons in the selection, an additional 2% of ψ(2S) candidates is selected, from which a feed-down of (2.0 ± 2.0)% is estimated.

Inelastic background determination
The largest background is due to diffractive J/ψ and ψ(2S) meson production with additional gluon radiation or proton dissociation (see Figs. 1(b),(c),(d)) where the particles produced go outside the LHCb acceptance and, in particular, close to the beam-line. Proton dissociation is more likely to occur when the transverse momentum of the meson is higher, while additional gluon radiation also increases the average p T . Thus the inelastic background has a higher p T than the signal. In Regge theory, it is assumed that for elastic J/ψ and ψ(2S) meson production T c 2 is the four-momentum transfer squared at the protonpomeron vertex. The H1 and ZEUS collaborations confirmed this dependence and measured b s = 4.88 ± 0.15 GeV −2 [8] and b s = 4.15 ± 0.05 +0. 30 −0.18 GeV −2 [9], respectively, for J/ψ production, while H1 measured b s = 4.3 ± 0.6 GeV −2 [13] for ψ(2S) production. In contrast, the proton dissociative production of J/ψ or ψ(2S) for |t| < 1.2 GeV 2 is observed to follow an exponential dependence, exp (b pd t), with b pd = 1.07 ± 0.11 GeV −2 for J/ψ and b pd = 0.59 ± 0.17 GeV −2 for ψ(2S) [27]. For larger values of |t| a power law is required [8].
The values of b measured at HERA can be extrapolated to LHC energies using Regge , with W 0 = 90 GeV and α = 0.164 ± 0.041 GeV −2 [7] for the elastic process while α = −0.014 ± 0.009 GeV −2 [27] for proton dissociation. This predicts b s ≈ 6 GeV −2 and b pd ≈ 1 GeV −2 in the LHCb kinematic region. After the non-resonant contribution has been subtracted using the side-bands indicated in Fig. 3, and with the requirement of p 2 T < 0.8 GeV 2 /c 2 for the J/ψ and ψ(2S) removed, the data are fitted to the function where f s and f pd are the fractions of elastic and proton-dissociative production, respectively, and f fd is the fraction of feed-down fixed to that obtained in Sec. 3.2. The shape of the distribution for the feed-down contribution, F fd , is taken from the data using χ c → J/ψ γ and ψ(2S) → J/ψ ππ candidates. The numbers N 1 , N 2 and N 3 normalise each of the three functions to unity in the region p 2 T < 0.8 GeV 2 /c 2 , while b s and b pd are free parameters. The result of the fit for the J/ψ sample is shown in Fig. 4(a). The χ 2 /ndf of the fit is 115/96 and returns values of b s = 5.70 ± 0.11 GeV −2 and b pd = 0.97 ± 0.04 GeV −2 . Below p 2 T = 0.8 GeV 2 /c 2 , the signal fraction is 0.597 ± 0.012 and correcting for the non-resonant contribution gives an overall purity for the J/ψ sample of 0.592 ± 0.012. The result of the fit for the ψ(2S) sample is shown in Fig. 4(b). The χ 2 /ndf of the fit is 11/16 and returns values of b s = 5.1 ± 0.7 GeV −2 and b pd = 0.8 ± 0.2 GeV −2 . Below p 2 T = 0.8 GeV 2 /c 2 , the signal fraction is 0.62 ± 0.08 and correcting for the non-resonant contribution gives an overall purity for the ψ(2S) sample of 0.52 ± 0.07. In both cases, the values obtained for b s and b pd are in agreement with the extrapolations of HERA results using Regge theory.
A systematic uncertainty is assigned due to the choice of the fit range and the shape of the parameterisation describing the inelastic background. Doubling the range of the fit for the ψ(2S) candidates changes the signal fraction by 3%. Doubling the range of the fit for the J/ψ candidates leads to a poor quality fit; a single exponential function does not describe the background well. For large values of p 2 T , the H1 collaboration introduced a function of the form (1 + b pd p 2 T /n) −n which interpolates between an exponential at low where the non-resonant background contribution has been subtracted using side-bands. The points are data, the solid curve is the total fit while the different contributions are as described.
p 2 T and a power law at high p 2 T [8]. Using this functional form and holding n = 3.58, as determined by H1, gives an acceptable fit with the signal fraction changing by 5%. This value is used as a systematic uncertainty on the purity determination for both the J/ψ and ψ(2S) analyses.
The calculated fraction of feed-down is found to have a negligible effect on the fraction of signal events as the shapes are similar to the fitted shape for the proton dissociation. The purity of the sample is also calculated in bins of rapidity. No trends are observed and all values are consistent within statistical uncertainties. Consequently a single value for the purity of each sample is assumed, independent of the rapidity of the meson.

Cross-section calculation
The cross-section times branching fraction to two muons with pseudorapidities between 2.0 and 4.5 is calculated in ten bins of meson rapidity, y, via where: N i is the number of events in bin i; ρ is the purity of the sample as described in the previous section; A i is the acceptance of the fiducial region as shown in Fig. 2; i is the efficiency to select a signal event; ∆y is the bin width; L is the luminosity, which has been determined with an uncertainty of 3.5% [28]; and single is the efficiency for selecting single interaction events, which accounts for the fact that the selection requirements reject signal events that are accompanied by a visible proton-proton interaction in the same beam crossing. The number of visible proton-proton interactions per beam crossing, n, is assumed to follow a Poisson distribution, P (n) = µ n exp(−µ)/n!, where µ is the average number of visible interactions, defined as interactions with one or more tracks having VELO information. The probability that a signal event is not rejected due to the presence of another visible interaction is given by P (0) and therefore, single = exp(−µ). This has been calculated throughout the data-taking period in roughly hour-long intervals; variations in µ during this interval have been studied and found to have a negligible effect. The spread in the value of µ for different crossing bunch-pairs is small and its effect is neglected. Spillover from previous beam crossings does not affect the VELO, and detector noise is typically at the level of a few uncorrelated hits, which do not form a track. Averaged over the data-taking period, single = 0.241 ± 0.003, where the uncertainty has been calculated with the assumption that at least one track in a visible interaction may be spurious.
For events with two tracks inside the fiducial region, the efficiency to select a meson can be expressed as the product ψ id × ψ trig × sel , where ψ id is the efficiency for both tracks from the meson to be identified as muons, ψ trig is the efficiency for an event with two reconstructed muons to fire the triggers, and sel is the efficiency for a triggered event to pass the selection criteria. Most of these quantities are calculated directly from the data.
For muons coming from J/ψ decays, a tag-and-probe technique is used to find the efficiency to identify a single muon, µ id (φ, η), as a function of the azimuthal angle, φ, and pseudorapidity, η. The tag is an identified muon that fires the trigger. The probe is the other track in events with precisely two tracks, which gives the J/ψ invariant mass when combined with the tag-track. The fraction of probes identified as muons determines µ id (φ, η) from which ψ id (y) is found using the simulation to relate the rapidity of the meson to the azimuthal angles and pseudorapidities of the two muons. A scaling factor (typically 1%), found using simulated events, is applied to get the corresponding quantity for muons from ψ(2S) decays, which have a slightly different momentum spectrum due to the higher mass of the ψ(2S) meson.
In a similar way, the tag-and-probe technique is used to find the efficiency for a single muon to fire the hardware trigger, from which is derived the corresponding J/ψ efficiency. The software trigger requires the presence of two muons. Its efficiency is determined using an independent pre-scaled software trigger that fires on a single muon. Combining these factors together gives a data-driven estimate of ψ trig for J/ψ decays. A correction of typically 3%, found using simulated events, is applied to get the corresponding quantity for ψ(2S) decays. The values determined for ψ id × ψ trig as a function of rapidity are given in Table 1.
The selection efficiency, sel , is determined using simulation and data. From simulation the requirement of having no additional tracks in the event has an efficiency of 0.997±0.001, and the requirement that p 2 T < 0.8 GeV 2 /c 2 has an efficiency of 0.979 ± 0.003, where the uncertainty corresponds to a change of 5% in the value of b s used in the simulation. The simulation, calibrated using a sample of J/ψ + γ candidates in data, determines the efficiency for having no identified photons in the event to be 0.972 ± 0.005. The requirement that the meson mass lies within 65 MeV of the known value is found from the fit in Fig. 3 and gives an efficiency of 0.96 ± 0.01. The global requirement, imposed by the hardware trigger, that there be fewer than ten SPD hits introduces a small inefficiency due to spillover from previous beam crossings. The efficiency of the SPD requirement is measured to be 0.96 ± 0.01 using an independent trigger that has no constraint on the number of SPD hits. Combining these together, it is estimated that sel = 0.87 ± 0.01 for the J/ψ selection and the same value is taken for the ψ(2S) analysis.
The numbers entering the cross-section calculation are summarised in Table 1. Applying Eq. 1 leads to differential cross-section times branching fraction values for mesons with both muons inside 2.0 < η µ ± < 4.5, which are reported in Table 2. The uncorrelated statistical uncertainties are combined in quadrature and are reported in the top half of the table. The statistical uncertainties on sel and the purity are correlated between bins and are quoted in the lower half of the table. The systematic uncertainties on the fraction of single interaction beam-crossings, the acceptance, the shape of the inelastic background and the luminosity are also correlated between bins and are indicated separately.
The total cross-section times branching fraction to two muons is obtained by integrating the cross-sections in Table 2. The uncorrelated statistical uncertainties are added in quadrature and combined with the correlated ones to give the total statistical uncertainty. The correlated systematic uncertainties, indicated in the lower part of Table 2 by asterisks,  are combined to give the total systematic uncertainty. This leads to the following results for the cross-section times branching fraction to two muons having pseudorapidities between 2.0 and 4.5: where the first uncertainty is statistical and the second is systematic. The differential cross-section results reported in Table 2 correspond to muons that enter the LHCb detector fiducial volume. Differential cross-section results as a function of meson rapidity are obtained by dividing by the known meson branching fraction to two muons [22] and by the fraction of decays with 2.0 < η µ ± < 4.5. The former introduces an 13.0% Total correlated systematic uncertainty 6.5% additional systematic uncertainty of 10% for the ψ(2S) measurement. The latter depends on the kinematics of the decay, is calculated using SuperChic assuming that the J/ψ and ψ(2S) mesons are transversely polarised, and is given in Table 3.  [29] 275 JMRT [5] 282 8.3 Motyka and Watt [2] 334 Schäfer and Szczurek [30] 317 Starlight [31] 292 6.1 Superchic [19] 317 7.0 LHCb measured value 291 ± 7 ± 19 6.5 ± 0.9 ± 0.4

Discussion
The integrated cross-section measurements for J/ψ and ψ(2S) mesons decaying to muons with 2.0 < η µ ± < 4.5 are compared to various theoretical predictions in Table 4. Good agreement is found in each case. The differential distribution for J/ψ production is presented in Fig. 5(a), where the extent of the error bars indicates the uncorrelated statistical uncertainties and the band is the total uncertainty. Jones, Martin, Ryskin and Teubner (JMRT) [5] have obtained leading-order (LO) and next-to-leading-order 1 (NLO) predictions from a fit to HERA and LHCb 2010 data, which are dominated by the HERA data; thus these curves can be considered as LO and NLO extrapolations from HERA energies. The LO result is essentially the power-law photoproduction result from HERA, combined with a photon flux function and a gap survival factor [32]. Better agreement is obtained between data and the NLO prediction than between data and the LO prediction.  Figure 6: Photoproduction cross-section as a function of the centre-of-mass of the photon-proton system with the power-law fit from [8] superimposed. The LHCb data points for W + (W − ) are derived assuming the power-law fit for W − (W + ). The uncertainties are correlated between bins. Fixed target results are from the E401 [33], E516 [34] and E687 [35] collaborations.
Exclusive production of J/ψ in pp collisions is related to photoproduction through where dn/dk ± are photon fluxes for photons of energy k ± ≈ (M J/ψ /2) exp(±|y|), (W ± ) 2 = 2k ± √ s, and r ± are absorptive corrections as given, for example, in [5,30]. The LHCb results cannot unambiguously determine the photoproduction cross-section due to contributions from both W + and W − , corresponding to the photon being either an emitter or a target, respectively. However, a comparison can be made to the HERA photoproduction results using the power-law relationship, σ γp→J/ψ p (W ) = 81(W/90 GeV) 0.67 nb, determined by the H1 collaboration [8]. A model-dependent measurement of σ γp→J/ψ p (W + ) is obtained from the LHCb differential cross-section measurement by applying Eq. 3 and assuming the power-law result for σ γp→J/ψ p (W − ), while σ γp→J/ψ p (W − ) is obtained by assuming the power-law result for σ γp→J/ψ p (W + ). The result of this procedure is shown in Fig. 6, which compares the modified LHCb data with HERA and fixed target photoproduction results: note that there are two correlated points plotted for each LHCb measurement, corresponding to the W + and W − solutions. It was shown in our previous publication [11] that the LHCb data were consistent, within large statistical uncertainties, with a simple power-law extrapolation of HERA J/ψ photoproduction results to LHC energies. With increased statistics, an extrapolation of the power-law obtained in [8] is in marginal agreement with the LHCb data. Ducati, Griep and Machado [3] and Motyka and Watt [2], which include saturation effects. The points are data where the error bars indicate the uncorrelated uncertainties. The band indicates the total uncertainty, most of which is correlated between bins.
The differential distribution for ψ(2S) production is presented in Fig. 5(b) and is compared to both LO and NLO predictions 2 from JMRT [36] using the formalism described in [5] with the gluon PDF taken from their J/ψ analysis. Once again, better agreement is found between data and the NLO prediction than between data and the LO prediction.
In addition to higher order effects being capable of explaining the deviation from a pure power-law behaviour, saturation effects may be important. Figure 7(a) compares the J/ψ differential distribution to predictions by Motyka and Watt [2] and Gay Ducati, Griep and Machado [3], that both include saturation effects and have a precision of 10 − 15%. A rapidity gap survival factor of r(y) = 0.85 − 0.1|y|/3 has been applied to the former while the latter assumes r(y) = 0.8. Both predictions use a Weizsäcker-Williams approximation to describe the photon flux. The agreement with the LHCb data is good. Figure 7(b) compares the ψ(2S) differential distribution to the prediction of Gay Ducati, Griep and Machado. Good agreement with the data is again observed.

Conclusions
The differential and integrated cross-section times branching fraction for J/ψ and ψ(2S) mesons decaying to two muons, both with 2.0 < η < 4.5, have been measured. The results of this analysis are consistent with the previously published LHCb analysis, which used data taken in 2010, but have a significantly improved precision, as well as a more extensive use of data-driven techniques to estimate systematic sources. An increase in luminosity, lower pile-up running conditions, as well as improvements in the trigger lead to roughly 40 times as many events in the 2011 data-taking period. The integrated cross-section measurements have an overall uncertainty that is a factor two better; they are limited by the theoretical modelling of the inelastic background for the J/ψ analysis and by the statistical precision with which the background is determined for the ψ(2S) analysis. The cross-section is presented differentially for the ψ(2S) for the first time. Although the total uncertainty for the J/ψ differential distribution is 7% per rapidity bin, most of this is correlated bin-to-bin; the uncorrelated uncertainty is typically 1.5%. Thus the overall shape of the differential distribution is rather well determined and this improves the ability of the data to distinguish between different theoretical models.
The integrated cross-sections are in good agreement with several theoretical estimates. The differential J/ψ and ψ(2S) cross-section both agree better with the NLO rather than LO predictions of [5]. The result has also been compared to two models that include saturation effects [2,3]; in both cases, good agreement is observed. It is also worth noting that the t-dependence for LHCb and HERA results are in agreement with Regge theory.