J/psi and psi(2S) production in pp collisions at sqrt(s) = 7 TeV

A measurement of the J/psi and psi(2S) production cross sections in pp collisions at sqrt(s)=7 TeV with the CMS experiment at the LHC is presented. The data sample corresponds to an integrated luminosity of 37 inverse picobarns. Using a fit to the invariant mass and decay length distributions, production cross sections have been measured separately for prompt and non-prompt charmonium states, as a function of the meson transverse momentum in several rapidity ranges. In addition, cross sections restricted to the acceptance of the CMS detector are given, which are not affected by the polarization of the charmonium states. The ratio of the differential production cross sections of the two states, where systematic uncertainties largely cancel, is also determined. The branching fraction of the inclusive B to psi(2S) X decay is extracted from the ratio of the non-prompt cross sections to be: BR(B to psi(2S) X) = (3.08 +/- 0.12(stat.+syst.) +/- 0.13(theor.) +/- 0.42(BR[PDG])) 10^-3


Introduction
Quarkonium production at hadron colliders provides important tests of calculations in the context of both perturbative and non-perturbative quantum chromodynamics (QCD), via measurements of production cross sections and polarizations.
The J/ψ and ψ(2S) mesons can be produced in proton-proton (pp) collisions through two mechanisms: prompt mesons directly produced in the primary interaction and non-prompt mesons from the decay of directly produced b hadrons. In addition, J/ψ production can also occur via decays of heavier charmonium states, both S-wave (the ψ(2S) itself) and P-wave (the three χ c states). The determination of the latter contribution is challenging both theoretically and experimentally, because it requires detection of the low-energy photons from χ c decays. In nonrelativistic QCD (NRQCD) models [1,2], by adding a contribution to prompt charmonium production through colour-octet states [3], a satisfactory description of prompt J/ψ and ψ(2S) meson cross sections at the Tevatron [4] has been obtained. However, in these calculations the fraction of J/ψ originating from χ c decays must be assumed from experimental measurements with large uncertainties, which makes the ψ(2S) mesons cleaner probes of NRQCD predictions.
Non-prompt J/ψ and ψ(2S) production can be directly related to b-hadron production, providing a measurement of the b-hadron cross section in pp collisions. Past discrepancies between the Tevatron results and the next-to-leading-order (NLO) QCD calculations have been resolved using the fixed-order next-to-leading-log (FONLL) approach and updated measurements of the b → J/ψ fragmentation and decay [5,6].
Measurements of the prompt and non-prompt production cross sections of J/ψ mesons decaying to muon pairs was published using the first Compact Muon Solenoid (CMS) data [7], as well as data from other LHC experiments [8][9][10]. The present analysis extends the CMS result with a larger amount of statistically independent data and complements it by providing a measurement of the ψ(2S) production cross section, as well as the ratio of the two cross sections. Higher trigger thresholds induced by the increased LHC luminosity do not allow the current measurement to reach charmonium transverse-momentum (p T ) values as low as in Ref. [7], but the high-p T reach is increased by the much larger amount of data. The advantage of measuring the ψ(2S) to J/ψ cross-section ratio lies in the cancellation of several experimental and theoretical uncertainties.
The polarizations of the J/ψ and ψ(2S) states affect the muon momentum spectrum in the laboratory frame, thus influencing the charmonium acceptance and, as a consequence, the extracted cross sections. Therefore it was decided to present the results in two different ways. The first approach assumes unpolarized production for prompt J/ψ and ψ(2S) whereas nonprompt mesons are assumed to have the polarization generated by the EVTGEN Monte Carlo program [11], corrected to match the most recent measurements [12]. Typical changes of the measured cross sections resulting from using hypotheses of full longitudinal or full transverse polarizations are also given. The second approach provides results restricted to the phase-space region of the CMS muon detector acceptance, in order to avoid corrections which depend on the unknown polarizations of the two charmonium states.
the leading-order colour-singlet and colour-octet mechanisms. Colour-octet states undergo a shower evolution. We use the NRQCD matrix element tuning obtained by fitting NRQCD calculations to CDF data [16,17]. In the absence of consistent theoretical and experimental information about the J/ψ and ψ(2S) polarizations, the dilepton decay distribution is assumed to be isotropic. Simulated events with b-hadron decays are also generated with PYTHIA, and the b hadrons are forced to decay inclusively into J/ψ and ψ(2S) using the EVTGEN package. Photon final-state radiation (FSR) is implemented using PHOTOS [18,19].
The off-line event selection, only briefly summarized here, is very similar to the one used in Ref. [7]. Muon candidates are reconstructed from the combination of muon-detector and silicon-tracker hits. The muons are required to pass the following criteria in the tracker: have at least 10 tracker hits, at least two of which are required to be in the pixel layers; have a χ 2 per degree of freedom smaller than 1.8; and pass within a cylinder of radius 3 cm and length 30 cm centred at the beam-spot centroid position and parallel to the beam line.
To select events with J/ψ or ψ(2S) decays, muons with opposite charge are paired and their invariant mass is computed. The mass is required to be between 2.5 and 4.7 GeV/c 2 . The two muon trajectories are refitted with a common vertex constraint, and events are retained if the χ 2 probability of the fit is larger than 1%. If more than one muon pair is found in the event, the one with the largest vertex χ 2 probability is retained.
The dimuon L1 triggers include a veto, whose specific criteria depend on the type of muon chamber and the region of the detector: this rejects muon signals whose spatial separation in the muon stations is too small, in order to avoid spurious dimuon signatures from a single muon. As a consequence, the dimuon sample is split in two, depending on the signed difference in azimuthal angle (∆φ) between the positively and the negatively charged muons. Muons that bend towards each other in the magnetic field are called type-C ("convergent") dimuons, while muons that bend away from each other are type-D ("divergent") dimuons.
Dimuons of type D are much less affected by the trigger veto, while dimuons of type C may cross at the muon stations. This causes sizeable correlations between the two muon detection efficiencies, and this effect is larger in the forward region. Therefore, in addition to the above requirements, all type-C dimuons are rejected for the inclusive cross-section measurements. This corresponds to a 48% reduction in the yield. For the non-prompt fraction determination (which is largely efficiency independent), type-C dimuons are only rejected in the high dimuon rapidity region 1.6 < |y| < 2.4.
The momentum measurement of charged tracks in the CMS detector has systematic uncertainties that are due to imperfect knowledge of the magnetic field, modelling of the detector material, and sub-detector misalignment. These effects can shift and change the width of the mass peaks of dimuon resonances. In addition to calibrations already applied to the data [20][21][22], residual effects are determined by studying the dependence of the reconstructed dimuon peak shapes on the muon kinematics, as was done in Ref. [7].
Because of the large difference in the branching fractions to dimuons of the two states, the measured ψ(2S) yield is much smaller than the J/ψ yield. For this reason, different binnings are used for the differential cross sections in p T and |y|.

Inclusive yield determination
Two methods are used to extract the inclusive yields from the µ + µ − invariant mass distribution, either fitting the J/ψ peak alone in a restricted mass window, or fitting the combined J/ψ and ψ(2S) distribution. Yields are derived using an extended unbinned maximum-likelihood method.
In both types of fits, the sum of a Gaussian and a Crystal Ball [23] function is used for the description of the signal, simultaneously taking into account FSR and rapidity-dependent resolution variations. In the J/ψ-only fits, an exponential function is used to describe the background. Figure 1 (top) shows an example of a fitted mass distribution. The mass resolution ranges from about 20 MeV/c 2 in the low rapidity region to 35 MeV/c 2 at intermediate rapidities, up to about 50 MeV/c 2 in the forward, high rapidity region.
In the second type of fits, the two mass peaks and the background are fitted simultaneously. For the J/ψ and ψ(2S) signal peaks, the same probability density function (pdf) is used, with the following constraints on the parameters: the ratio of the central values of the two masses is fixed to the world average value [24]; the widths, scaled by the nominal mass values, are constrained to be the same, and the parameters describing the asymmetric tail of the Crystal Ball function are constrained to be equal. The background is modelled by two exponentials. Figure 1 (bottom) shows an example of a fitted mass distribution.  Figure 1: Top: The µ + µ − invariant mass distribution in the J/ψ region and the result of the fit for the bin: 0.9 < |y| < 1.2, 12 < p T < 15 GeV/c. Bottom: J/ψ and ψ(2S) mass distribution fit for the bin: |y| < 1.2, 8 < p T < 9 GeV/c. The solid and dashed lines represent the total fits and their background components, respectively.
The systematic uncertainties on the mass distribution fits are estimated by changing the analytical form of the signal and background pdf hypotheses (a single Crystal Ball function is used for the signal and polynomial pdfs for the background) in the two types of fits. The largest variation in the yield of each fit is taken as the systematic uncertainty.

Acceptance and efficiency
As discussed in Section 1, measurements will be presented using two different approaches. In the first approach, the observed number of J/ψ events is corrected for the detector acceptance and reconstruction efficiency in every bin in which the cross section is measured. As the acceptance is strongly dependent on the assumed polarization of the charmonium state, in the second approach we provide measurements exclusively within the CMS detector acceptance, where only detector efficiency corrections are made and without any polarization-related uncertainties.

Acceptance
The acceptance reflects the geometrical coverage of the CMS detector and the kinematic reach of the muon trigger and reconstruction, constrained by the amount of material in front of the muon detectors and by the track curvature in the magnetic field.
In the simulation, both muons are required to be within the geometric acceptance of the muon detectors. A single muon is defined as detectable if it satisfies the following requirements at generator level: The J/ψ acceptance A is defined as the fraction of detectable J/ψ → µ + µ − decays, as a function of the generated dimuon transverse momentum p T and rapidity y, where N det is the number of detectable J/ψ events in a given (p T , y) bin, and N gen is the corresponding total number of generated J/ψ events in the Monte Carlo (MC) simulation. An analogous definition holds for ψ(2S). The parameter λ θ reflects the fact that the acceptance is computed for various polarization scenarios, which lead to different muon spectra in the laboratory frame.
For the acceptance calculation, a dedicated sample of generated events is used, with no restrictions on the phase space. The large number of simulated events allows a much smaller bin size for determining A with respect to that used for the cross-section determination in data.
To study the effect of J/ψ and ψ(2S) polarization on the acceptance, these events are reweighted depending on the values of the polar and azimuthal angles as computed in two different frames (helicity and Collins-Soper [25]). The angular distribution for the decay of a J = 1 state into fermions is used, which is a function of three independent parameters λ θ , λ φ , and λ θφ : · (1 + λ θ cos 2 θ + λ φ sin 2 θ cos 2φ + λ θφ sin 2θ cos φ) .

Acceptance and efficiency
The choice of zero for all λ parameters corresponds to an unpolarized decay, while λ θ = −1 and λ θ = +1 correspond to fully longitudinal and fully transverse polarizations, respectively.
By default, the prompt J/ψ and ψ(2S) are assumed to be unpolarized, while the non-prompt mesons are assumed to be polarized as generated by EVTGEN and corrected to match recent measurements, as mentioned in the Introduction. Typical changes of the measured cross sections when using alternative polarization scenarios are provided in Section 8.
Several sources of systematic uncertainty on the acceptance have been investigated: • Kinematic spectra. Different p T and y spectra of the generated J/ψ and ψ(2S) might produce different acceptances, as the acceptance is defined by single-muon criteria. Spectra from theoretical predictions presented in Section 8 have been used to recompute the acceptance, and the difference from that obtained with the PYTHIA spectrum has been taken as a systematic uncertainty.
• Final-state radiation. The generated dimuon momentum may differ from the J/ψ and ψ(2S) momentum, because of FSR. The difference between the acceptance computed using the dimuon or the charmonium variables in Eq. (2) is taken as a systematic uncertainty.
• B polarization. The J/ψ and ψ(2S) mesons produced in b-hadron decays have a different acceptance with respect to the prompt ones. The corresponding systematic uncertainty is evaluated by taking the difference of the default choice (corrected to match the experimental results of Ref. [12]) with respect to the one predicted by EVTGEN.
• p T calibration. The muon transverse momenta in data have been calibrated as described in Ref. [7]. A difference in the momentum resolution between data and simulated events would also give a systematic uncertainty on the acceptance. The acceptance has been computed with simulated muon momenta smeared according to the resolution measured with data [26]. The uncertainty on the measured resolution was used to apply an additional smearing on the simulated momenta; the acceptance has been recalculated and the shift taken as a systematic uncertainty.

Muon efficiency
The single-muon efficiency is measured from data for muons in the acceptance, as described in Refs. [27,28], and is based on the tag-and-probe (T&P) method. For this purpose, independent sets of triggers are used for which online requirements either on the muon or the tracker tracks are not applied, thus yielding samples which are unbiased with respect to the corresponding selections.
The combined trigger and offline reconstruction efficiency for a single muon is defined as: where track is the offline tracking efficiency, ID | track refers to the muon identification in the muon systems for a tracker-reconstructed muon, off | ID refers to the specific quality requirements applied to reconstructed muons, and trig | off is the probability for an offline reconstructed muon to have also fired the trigger.
The muon identification and trigger efficiencies ( trig | off , off | ID and ID | track ) have the strongest p µ T and |η µ | dependence and are determined in 15 bins of p µ T (3.3 < p µ T < 50 GeV/c) and 14 bins of |η µ | (0 < |η µ | < 2.4), allowing an adequate description of the turn-on efficiency curves. Since the tracking efficiency is almost constant for this momentum and rapidity range, broader bins are used.
The efficiency to detect a dimuon event is expressed as: where (µ + ) and (µ − ) are the single-muon efficiencies, and vertex is the efficiency of the vertex χ 2 requirement, calculated from the data by determining the yields in regions of large p T and |y| by alternatively applying and not applying this requirement. The ρ factor, defined by Eq. (5), represents a correction to the efficiency factorization hypothesis: it accounts for the finite size of the (p µ T , η µ ) bins and, more importantly, for the possible bias introduced by the T&P measurement, due to correlation effects as discussed in Section 3. In order to determine ρ, the efficiencies have also been evaluated using T&P techniques on simulated events and their product has been compared with the true dimuon efficiency. Except for some bins at high p T , the values are found to satisfy |1 − ρ| < 10%.
For the acceptance-corrected cross-section results, the acceptance and efficiencies are combined into a single factor, which is computed for each (p T , y) bin and is defined as: where the average is taken over the data events in each bin, using the "fine-grained" bins of the acceptance and the event-by-event efficiency obtained from the single-muon efficiencies using Eq. (5).
Similarly, for the results which are not corrected for acceptance, the efficiency factor is determined as: Two sources of systematic uncertainty in the efficiency are considered: • The uncertainties on the measured muon efficiencies propagate as systematic errors on the cross section measurement through the correction factor 1 A· bin (or 1 bin ). The effect has been estimated on a statistical basis in each bin by performing Monte Carlo pseudo-experiments, in which the muon efficiencies were varied randomly according to a probability density built by joining the left and the right side of two Gaussians with different widths, in order to allow for asymmetric errors. The r.m.s. of these correction factors in each bin has been taken as the systematic uncertainty associated with the single-muon efficiency.
• The full difference |1 − ρ| in each bin is taken as a systematic uncertainty due to the efficiency correlation.

Inclusive cross-section determination
The inclusive double-differential cross section is given by: where L dt is the integrated luminosity, ∆p T and ∆y are the p T and y bin widths, B(J/ψ → µ + µ − ) is the decay branching fraction of the J/ψ into two muons, and N corr J/ψ (p T , |y|) is the corrected J/ψ yield in a given (p T , |y|) bin. The corrected yield is obtained from the fitted signal yield N J/ψ via N corr J/ψ = N J/ψ · 1 A· bin in the case where the J/ψ yields are corrected for acceptance and efficiency, and N corr J/ψ = N J/ψ · 1 bin in the case where the results are uncorrected for acceptance. An analogous formula applies for the ψ(2S) double-differential cross section. Figure 2 shows the measured (fully corrected) J/ψ inclusive cross section as a function of p T for the various rapidity bins. They are compared with our previous results published in Ref. [7], which are statistically independent and remain of interest since they partially overlap with the present results and cover a lower p T range. A good agreement is observed. In this figure, as well as in the cross-section plots of Section 8, multiplicative factors -appearing as additive offsets on the log scale -are used to achieve a convenient graphical separation of the measurements from different rapidity bins.  2: Measured differential cross section for J/ψ inclusive production as a function of p T for five rapidity bins, fully corrected for acceptance and efficiency. Also plotted are the results published in Ref. [7], which extend to a lower p T range. The error bars on data points include all the statistical and systematic contributions except luminosity and polarization. The measurements have been offset by the numerical values given in the legend for easier viewing.

Prompt and non-prompt fractions
To estimate the J/ψ fraction from b-hadron decays, a two-dimensional fit is performed, in which the pdfs and fit procedure are the same as those described in Ref. [7]. The variables used for the two-dimensional fits are the dimuon invariant mass and the "pseudo proper decay length" J/ψ , defined as the most probable value of the transverse distance between the dimuon vertex and the primary vertex, corrected by the transverse Lorentz boost of the J/ψ. As in Ref. [7], the primary vertex is chosen as the one closest to the dimuon vertex in the z direction.
The resolution of the pseudo proper decay length is described by a function depending on an event-by-event uncertainty determined from the covariance matrices of the primary and secondary vertex fits. The uncertainty is used as the r.m.s. of the resolution Gaussian function that describes the core of the resolution, while a second Gaussian function with a small relative normalization (usually < 1%) parametrizes the effect of incorrect primary vertex assignments.
The pdf F( J/ψ , m µµ , σ ) for the J/ψ is then: where: and k = {Sig, Bkg}. In the equations above: • M Sig (m µµ ) and M Bkg (m µµ ) are the mass pdfs determined for the signal and background in Section 4, and f Sig is the fraction of signal events in the entire range of the fit; • F true Sig ( J/ψ ) and F true Bkg ( J/ψ ) are the functional forms describing the J/ψ distribution for the signal and background, respectively. The signal part is given by the sum of prompt and non-prompt components: is the fraction of J/ψ from b-hadron decays, and F p ( J/ψ ) and f b ( J/ψ ) are the J/ψ distributions for prompt and non-prompt J/ψ, respectively. The J/ψ pdfs for prompt signal and background are the same as in Ref. [7]. The non-prompt lifetime function is described by an exponential decay of the b hadron, with a Gaussian smearing function that accounts for the difference between the measured pseudo proper decay length and the proper decay length of the b hadron; • σ is the per-event uncertainty of the decay length and D Sig (σ ) and D Bkg (σ ) are its distributions separately for signal and background [29]. They are obtained from the signal region of the invariant mass distribution, after a sideband subtraction, and the sideband regions, respectively; • R 1 and R 2 represent the core and tail decay-length resolution Gaussian functions: µ is their common mean and s i represent scale factors for the per-event uncertainty, which are both left free in the fit to account for initial assumptions on the uncertainties of track parameters. These functions are convolved with F true k ( J/ψ ) to obtain the observed F k ( J/ψ ) distributions, including the experimental resolution (k = {Sig, Bkg}).
The background is fitted using the events in mass sidebands and the result is used to fix lifetime parameters of the overall fit in the entire mass region. The mass sideband region is defined as For the determination of the ψ(2S) non-prompt fraction, the quantity ψ(2S) , defined as for the J/ψ case, is computed. In order to constrain the fit and avoid problems due to limited statistical accuracy, the J/ψ and ψ(2S) samples are fitted simultaneously using the same binning as for the ψ(2S) cross-section determination. The lifetime resolution functions R 1 and R 2 are constrained to be described by the same parameters (mean value and scale factors) and the backgrounds to have the same fractions of long-lived components.
The invariant mass sideband regions used for the determination of the background parameters are defined as above for the J/ψ, and as [3.35, 3.45] and [3.85, 4.20] GeV/c 2 for the ψ(2S). Figure 3 shows two examples of the J/ψ and ψ(2S) distributions with projections of two-dimensional fits on these dimensions, as well as the prompt and non-prompt components obtained as described above. The lower plots in Figure 3 give the pull distributions from the fits, and show no systematic structure. Several sources of systematic uncertainty have been addressed, using mostly the same procedures as in Ref. [7]. The main additional systematic effect comes from attempting to choose the correct primary vertex of the interaction in the presence of pile-up. The sources of systematic uncertainty include the following: • Primary vertex assignment. In order to estimate the possible effect of pile-up on the primary vertex estimation, the primary vertex associated to the dimuon is chosen as the one with the largest track ∑ p T 2 , instead of the one closest in z to the dimuon vertex. The difference between the fitted non-prompt fractions in these two approaches is taken as the systematic uncertainty.
• Residual misalignment in the tracker. The effect of uncertainties in the measured misalignment of the tracker modules is estimated by reconstructing the data using different sets of alignment constants. The largest difference in the fit results with respect to the nominal case is taken as the systematic uncertainty.
• b-hadron lifetime model. An alternative fit method is used, namely the b-hadron lifetime model used in Ref. [7], which is based on MC templates; the difference in the fitted non-prompt fraction is taken as the systematic uncertainty.
• Background fits. The effect of a ±100 MeV/c 2 variation in the lower limit of the lowmass side (upper limit of the high-mass side) of the J/ψ (ψ(2S)) sideband boundaries is taken as the systematic uncertainty.
• Resolution model. The nominal (double-Gaussian) model for the pseudo proper decay length per-event resolution is compared with a model using a single-Gaussian shape. The difference is taken as the systematic uncertainty.
• Different prompt and non-prompt efficiencies. The MC simulation predicts slight differences between the prompt and non-prompt J/ψ and ψ(2S) efficiencies, mostly because of the different track densities from fragmentation products around the muons. These are taken into account; the relative difference is propagated to the non-prompt fraction, and taken as the systematic uncertainty.
Non-prompt fraction results are given in Section 8 and a summary of all the systematic uncertainties is given in Tables 1 and 2 for the J/ψ and ψ(2S), respectively.

Results
The prompt and non-prompt double differential cross sections for the two charmonium states are obtained by multiplying the measured inclusive cross sections with the fraction of prompt and non-prompt events, respectively. In addition the cross-section ratio of the two charmonium states is calculated.
Statistical uncertainties and contributions from the investigated sources to the total systematic uncertainties on these cross sections are summarized in Tables 1 and 2. The largest uncertainties are due to the efficiency correlations; FSR estimation has a sizeable effect only in bins close to the edges of the acceptance. They are compared with theoretical predictions from NRQCD [3] and from FONLL [5,6] for the prompt and non-prompt cases, respectively. Numerical values are also reported in Ref. [30].

Prompt and non-prompt cross sections corrected for acceptance
The NRQCD prediction includes non-prompt production in the J/ψ case caused by feed-down decays from heavier charmonia, and can therefore be directly compared with the data. Good agreement is found in both the J/ψ and the ψ(2S) cases. For non-prompt production the measurements lie systematically below the FONLL predictions, possibly because of the large uncertainty on the B → ψ(2S)X branching fraction. In general, for both states, the observed differential cross sections seem to fall more rapidly than the FONLL prediction at high p T , and this effect is more evident for the J/ψ because of the higher p T reach.
The NRQCD theoretical uncertainties include those on the feed-down contributions and on the colour-octet, long-distance matrix elements determined from fits to the Tevatron data. The Table 1: Summary of the relative statistical and systematic uncertainties on the non-prompt J/ψ cross section (in %). The variation over the different p T bins is given for the five rapidity regions. Uncertainties on the prompt cross section are identical, with the exception of the non-prompt fraction, where they must be regarded as relative to (1 − f b ) rather than to f b . Acceptance uncertainties on the FSR are given, excluding the lowest-p T bin in every rapidity region, where it can be as large as 19% because of acceptance edge effects. However, uncertainties on the B → charmonium decay spectrum were not included in the original FONLL prediction. To estimate those, we make use of the EVTGEN MC generator, which describes B → charmonium decays using a sum of many exclusive modes. We split the decay modes into two categories, "high-Q" and "low-Q", if the value of Q in the decay is respectively greater than or less than 1.2 GeV/c 2 , where Q = m B − ∑ i m i and the index i runs over the B decay products. As low-Q (high-Q) modes yield charmonia with smaller (larger) momentum in the Brest frame, they populate different regions of the B-decay spectrum. Two sets of nonprompt charmonium MC events are generated according to the following criteria. In the first, each high-Q mode branching fraction is increased by its world-average uncertainty [24] or by 100% of its value if the branching fraction is not measured. Low-Q mode branching fractions are decreased by a similar amount, rescaling the sum to unity after this procedure. In the second, the treatment of the high-and low-Q modes is interchanged. The maximum difference in the resulting spectra in the two cases is added to the theoretical FONLL uncertainty.
To investigate the effect of the assumed J/ψ and ψ(2S) polarizations on the prompt cross section, the acceptance is recomputed for four extreme polarization scenarios corresponding to fully longitudinal or fully transverse polarization in the helicity and Collins-Soper frames [25]. This produces relative cross-section shifts across the entire kinematic range of up to 18-20% (20-25%) for the J/ψ (ψ(2S)) in the helicity frame, and 6-15% for both states in the Collins-Soper frame. Detailed results can be found in Ref. [30]. Table 2: Summary of the relative statistical and systematic uncertainties in the non-prompt ψ(2S) cross section (in %). The variation over the different p T bins is given for the three rapidity regions. Uncertainties on the prompt cross section are identical, with the exception of the non-prompt fraction, where they must be regarded as relative to (1 − f b ) rather than to f b . Acceptance uncertainties on the FSR are given excluding the lowest-p T bin in every rapidity region, where it can be as large as 29% because of acceptance edge effects.

Prompt and non-prompt cross sections uncorrected for acceptance
As discussed previously, since the polarization effects are large compared to the measurement uncertainties, cross-section values are also reported that are restricted to the CMS muon acceptance region, to allow future measurements of the J/ψ and ψ(2S) polarization to be exploited. Figures 6 and 7 show the measured prompt and non-prompt cross sections for the J/ψ and the ψ(2S) as a function of p T for the various rapidity bins and uncorrected for detector acceptance. Numerical values can be found in Ref. [30].

Non-prompt fractions
The measured non-prompt fractions for J/ψ and ψ(2S) mesons, extracted as described in Section 7 and uncorrected for acceptance, are reported in Ref. [30] and shown in Fig. 8. The uncertainties shown are statistical and systematic, and the measured values are plotted as a function of p T in three rapidity ranges. In agreement with previous measurements [4,7], we observe similar sizes of non-prompt fractions for J/ψ and ψ(2S), and an increasing trend with p T . Acceptance corrections do not induce significant changes in the non-prompt fractions within their uncertainties.

Cross-section ratio
Most of the systematic uncertainties on the acceptances and efficiencies listed in Tables 1 and 2, as well as the luminosity uncertainty, cancel partially or fully in the ratio of the ψ(2S) to J/ψ cross sections. For this reason we also present the ratio of the two differential cross sections:      Figure 8: Fitted J/ψ and ψ(2S) non-prompt fractions plotted as a function of p T for three rapidity regions: 0 < |y| < 1.2 (top); 1.2 < |y| < 1.6 (middle); 1.6 < |y| < 2.4 (bottom). The inner error bars represent the statistical uncertainties only, the outer ones are the quadratic sum of statistical and systematic uncertainties.
where the ratio R is computed in bins of p T and rapidity, and the binning is the same as used for the ψ(2S) cross section.
The statistical uncertainties affecting R are extracted directly from the simultaneous invariant mass fits. The systematic uncertainties are estimated by considering the same sources as for the cross sections (except the luminosity and single-muon efficiency, which cancel out in the ratio) and evaluating directly the variation of the ratio, in order to take correlations into account.
No significant dependence of R on rapidity is observed; the ratios over the entire rapidity range are therefore computed. The resulting prompt and non-prompt cross-section ratios are shown  Figure 9: Measured value of R, the ψ(2S) to J/ψ differential cross-section ratio defined in Eq. 11, for prompt (left) and non-prompt (right) production, averaged over rapidity and plotted as a function of p T . The left plot also includes the comparison with the NRQCD prediction, while the right plot shows the predictions of the theoretical models used to determine B(B → ψ(2S)X), after the latter have been rescaled to the fitted value given in Eq. (12). The shaded bands show the uncertainties on the theoretical predictions. The error bars give the total uncertainties on the measurements; polarization uncertainties are not included.
in Fig. 9 as a function of p T . Numerical values of rapidity-dependent and integrated ratios are given in Ref. [30].
The assumptions on polarization also affect the prompt cross-section ratio measurement. In a plausible scenario [31], the polarizations of the directly produced J/ψ and ψ(2S) states are assumed to be the same. Therefore the uncertainty on the ratio comes only from the difference between the polarization of the directly produced mesons and the polarization of the J/ψ coming from decays of P-wave states (χ c1 and χ c2 ), for which the maximum possible variations are considered. Using the measured feed-down fractions measured at CDF [32,33], this leads to the definition of the two extreme scenarios: which result in changes to the measured prompt cross-section ratio by 12 − 20%.

Inclusive B → ψ(2S)X branching fraction
The non-prompt ψ(2S) cross-section results can be used to determine B(B → ψ(2S)X), the average inclusive branching fraction of all weakly decaying particles containing a bquark to ψ(2S).
Since the results are determined only for a limited range of phase space, a theoretical assumption is needed to extrapolate to the full phase space. The most precise result is obtained using the non-prompt cross-section ratio, where most theoretical uncertainties cancel. The FONLL model is used for the result, taking as an alternative the EVTGEN prediction to determine a systematic uncertainty.
For both models, the predicted ratio is computed for each p T bin used in the measurement, assuming the world-average values, listed by the Particle Data Group (PDG) [24], of B PDG (B → J/ψ X), B PDG (ψ(2S) → µ + µ − ), and B PDG (J/ψ → µ + µ − ) (the branching fractions taken from the PDG are indicated as B PDG ). In both models the production cross section for each type of B meson or baryon is weighted using the values of the fractions , taken from LEP and Tevatron measurements. The predictions are then fitted to the data points, leaving only the normalization (N) as a free parameter. A good agreement in the shape of the p T distribution is found for both models. The branching fraction B(B → ψ(2S)X) is then derived from the fitted normalization.
In addition to the fit uncertainty, including statistical and systematic uncertainties on the single measurements (±3.8%), the following sources of uncertainty are considered (with the corresponding relative ∆B uncertainty given in parentheses): • PDG branching fractions. The uncertainties quoted by the PDG for B(B → J/ψ X), B(ψ(2S) → µ + µ − ), and B(J/ψ → µ + µ − ) are summed in quadrature (13.5%). • Model assumptions. The EVTGEN prediction is used for an alternative fit and the difference with respect to the nominal N value is taken as a systematic uncertainty (1.0%).
• FONLL uncertainties. All uncertainties on the underlying bb cross section, discussed in Section 8.1, are assumed to be fully correlated between B → J/ψ X and B → ψ(2S)X transitions. Residual uncertainties affecting the cross-section ratio prediction are used to perform alternative fits and the differences with respect to the nominal value of N are taken as systematic uncertainties (1.5%).
• B → charmonium spectrum theoretical uncertainties. The only source of theoretical uncertainty which is not correlated is the one on the B → charmonium spectrum. We use the high-Q and low-Q method to estimate this uncertainty as detailed in Section 8.1. In order to obtain an upper limit on the uncertainty of this ratio, the high-Q sample of B → J/ψX is compared to the low-Q B → ψ(2S)X sample and vice versa. The average difference in N with respect to the nominal EVTGEN prediction is taken as a systematic uncertainty (3.8%).

Summary
A measurement of the J/ψ and ψ(2S) production cross sections in pp collisions at √ s = 7 TeV with the CMS experiment at the LHC has been presented. The data sample corresponds to an integrated luminosity of 36.7 ± 1.5 pb −1 . The two cross sections and their ratio have been measured as a function of the meson transverse momentum (up to 70 GeV/c for the J/ψ and to 30 GeV/c for the ψ(2S)) in several rapidity ranges. Cross sections for prompt and non-prompt production have been determined from the measured values of the J/ψ and ψ(2S) non-prompt fractions.
The prompt cross section results are evaluated assuming isotropic decays in the production, as well as four other polarization scenarios. In addition, cross sections restricted to the acceptance of the CMS detector are given, which are not affected by the polarization of the charmonium states.
Cross sections for prompt and non-prompt production have been compared with NRQCD and FONLL predictions, respectively. Agreement is found in the prompt case: this is particularly remarkable in the ψ(2S) case, where theoretical uncertainties are reduced because of the absence of feed-down from heavier charmonium states. In the non-prompt case, general agreement in shape is found for ψ(2S) in the entire p T range considered (up to 30 GeV/c), but an overall scale discrepancy is observed, possibly because of the assumption on the inclusive branching fraction B(B → ψ(2S)X). For J/ψ there is similarly general agreement over the above range, while the predictions overestimate the measured differential cross-sections for 30 < p T < 70 GeV/c.
For plausible hypotheses on the polarizations of the two charmonium states the ratio of their differential cross sections is obtained. In this ratio systematic errors largely cancel. The inclusive branching fraction B(B → ψ(2S)X) is extracted from the ratio of the non-prompt cross sections to be: B(B → ψ(2S)X) = (3.08 ± 0.12 (stat.+syst.) ± 0.13 (theor.) ± 0.42 (B PDG )) × 10 −3 , improving the relative uncertainty on the previous world average by a factor of three.